Device components formed of geometric structures

ABSTRACT

Various embodiments are directed to an apparatus and methods of forming and/or using an apparatus comprising a plurality of device components. An example method includes geometrically optimizing a periodic or aperiodic device comprising a plurality of device components by optimizing a topology, for each device component, from a starting point to have particular optical properties for a particular optical response. Each device component includes a plurality of geometric structures. The optimization includes selecting the starting point for a continuous profile to have the particular optical properties for the particular optical response, iteratively converging the continuous profile to a discrete profile, and, while iteratively converging to the discrete profile, adjusting edges between boundaries of the device components by accounting for fabrication constraints.

FEDERALLY-SPONSORED RESEARCH AND DEVELOPMENT

This invention was made with Government support under contractFA9550-15-1-0161 awarded by the U.S. Air Force. The U.S. Government hascertain rights in this invention.

OVERVIEW

Optics are used in a wide-variety of applications. For example, smartphones that are carried by people include optics or optical elements,such as lenses used in cameras. Due to the size of smart phones, as wellas other devices, the optics (e.g., hardware) have limits in size.Optical hardware processes spectral, polarization, and optical phaseinformation from incident light. Metasurfaces are sometimes used inoptics. Metasurfaces are optical hardware/devices that control theirmagnitude and phase response to light based on geometric design of themetasurfaces. For example, a metasurface controls the wavefronts ofincident electromagnetic waves and support beam steering and focusingfunctionality. Metasurfaces can also specify the polarization andangular momentum of light that exceeds the ability of conventionaloptical components.

Two factors affect metasurface design. The first is the detailedgeometry of geometric structures, which are sub-wavelength-scaleresonators with geometry-dependent optical properties. The second is thematerial forming the geometric structures, which is sometimes determinedby the target wavelength of interest. In infrared and terahertz regimes,metal and doped semiconductors are used, as well as plasmonic materialin both active and passive metasurface platforms. Transition metaloxides and nitrides can support plasmonic behavior in the near-infraredregimes. Between near-infrared and green visible light, semiconductorswith high refractive index and low loss, such as amorphous andpolycrystalline silicon, are used.

In accordance with various embodiments, silicon, such as crystallinesilicon or polycrystalline silicon and/or other various materials, isused to form geometric structures for a metasurface design. Crystallinesilicon, for example, has superior optical properties for bluewavelengths (400 nm to 500 nm range) as compared to polycrystallinesilicon or amorphous forms of silicon. For example, while amorphous orpolycrystalline silicon scatter light effectively at longer wavelengthsthan blue wavelengths, they absorb blue wavelengths which limits theirscattering efficiency. Aluminum is plasmonic at ultraviolet and bluewavelengths but its performance is limited by absorption losses andsensitivity to oxidation. Transparent dielectric materials such astitanium oxide and silicon nitride have low reflect indexes which limittheir ability to scatter light.

A number of aspects of the present disclosure are directed to methodsfor optimizing a thin film layer, such as a silicon-based metasurface.First, the metasurface layout is set (e.g., optimized) for a continuousprofile (e.g., continuous dielectric constant), and then the continuousprofile is converted into a discrete profile (e.g., realistic dielectricconstants). For example, the method includes optimizing a periodic oraperiodic device comprising a plurality of device components byoptimizing a topology, for each device component, from a starting pointto have particular optical properties for a particular optical response.Each device component includes a plurality of geometric structures. Theoptimization includes selecting the starting point for a continuousprofile to have the particular optical properties for the particularoptical response, converging the continuous profile to a discreteprofile, and adjusting edges between boundaries of the device componentsby accounting for fabrication constraints.

As further described herein, these devices are created using anadjoint-based topology optimization process, and possess non-intuitivelayouts that enable a variety of diffractive optics phenomena, such asbeam deflection and diffraction. The initial device component consistsof a random dielectric continuum of dielectric constants, with valuesranging between the dielectric constants of the material forming thegeometric structures, such as of air and silicon. To improve the Figureof Merit (FoM), which corresponds to grating efficiency, an iterativeprocess is performed that uses two electromagnetic simulations periteration, a forward and an adjoint simulation. These simulationsproduce two sets of electromagnetic field profiles within the device,which serve to simulate and/or specify specific changes in thedielectric constant at each location in a manner that improves the FoM.Over the course of multiple iterations, the dielectric continuum in thedevice converges to the dielectric constant of either silicon or air.The optimization method can be used to achieve multiple inputpolarizations and wavelengths, by performing forward and adjointsimulations for each optical degree of freedom per iteration.

Various aspects in accordance with the present disclosure includeperiodic or aperiodic metasurfaces that are formed by a plurality ofdevice components. Each device component includes multiple layers ofgeometric structures. For example, at least two layers includes a uniquegeometric layout of the geometric structures (e.g., nanostructures). Invarious aspects, one or more layers includes solid material, such as aSiO2 spacer layer that can be formed entirely of SiO2. The devicecomponents have a particular optical properties for a particular opticalresponse (e.g., are optimized to). For example, a device component isconfigured to control the amplitude and phase of light across abroadband spectrum, in various aspects. The geometric structures, insome specific aspects, are formed of silicon, such as crystallinesilicon that is capable of scattering light effectively across thebroadband spectrum (e.g., visible light, infrared and near-infraredlight). For example, various aspects of the present disclosure include ametasurface formed using multiple materials, wherein silicon and silicondioxide are the materials used to build at least a portion of themultiple layers of the metasurface (e.g., most), however, themetasurface is not limited to these materials.

According to other embodiments, methods and/or apparatuses (e.g.,devices, elements, and/or systems) are directed to a plurality of devicecomponents, each including at least one layer of geometric structures(e.g., each geometric structure including or being a material having ageometric shape and/or size defined by same-wavelength and/orsub-wavelength dimension(s)) and having optical properties for aparticular optical response, wherein the device components are combinedtogether to form a periodic or aperiodic device and/or apparatus. Inmore specific but related embodiments, the method and/or apparatus isfurther directed to a periodic or aperiodic device and/or apparatusincluding device components, each including at least two layers ofgeometric structures and having optical properties for a particularoptical response. Further, portions of the device components arecombined. For example, the device components can be combined together inat least two directions including width, length, and depth, and whereinthe stitched device components are used to manipulate light defined in aparticular wavelength range and wherein the shapes and/or sizes of thegeometric structures facilitate the manipulation of the light.

These and other matters have presented challenges to optics, for avariety of applications. The above discussion/summary is not intended todescribe each embodiment or every implementation of the presentdisclosure. The figures and detailed description that follow alsoexemplify various embodiments.

DESCRIPTION OF THE FIGURES

The patent or application file contains at least one drawing executed incolor. Copies of this patent or patent application publication withcolor drawing(s) will be provided by the Office upon request and paymentof the necessary fee.

Various example embodiments may be more completely understood inconsideration of the following detailed description in connection withthe accompanying drawings in the Appendices, which form part of thispatent document.

FIG. 1A illustrates an example geometric structure, in accordance withvarious embodiments;

FIG. 1B illustrates an example of a device component, in accordance withvarious embodiments;

FIG. 1C illustrates an example apparatus, in accordance with variousembodiments of the present disclosure;

FIGS. 2A-2C illustrates an example of an optimization process, inaccordance with various embodiments of the present disclosure;

FIGS. 3A-3C illustrate an example of a continuous profile and a discreteprofile of a topology of a device component, in accordance with variousembodiments;

FIGS. 4A-4D illustrate example boundary optimization of devicecomponents containing two geometric structures and three geometricstructures, in accordance with various embodiments;

FIG. 5 illustrates an example of topology and boundary optimization of adevice component with two geometric structures, in accordance withvarious embodiments;

FIGS. 6A-6B illustrate an example boundary optimization of a devicecomponent that is optimized to deflect light at a 20 degree angle, inaccordance with various embodiments;

FIGS. 7A-7D illustrate an example optimization of a device componentthat is optimized to deflect light at a 20 degree angle, in accordancewith various embodiments;

FIGS. 8A-8D illustrate an example device component, such as the devicecomponent illustrated by FIG. 1A, optimized for broadband spectrumtransmission efficiency, in accordance with various embodiments;

FIGS. 9A-9B illustrate an example device component, optimized forbroadband spectrum reflection efficiency, in accordance with variousembodiments;

FIG. 9C illustrate the relative transmission efficiency and absolutetransmission efficiency of light of two polarization states (e.g.,traverse electric polarization and transverse magnetic polarization) forthe polarization-insensitive device components illustrated by FIG. 1A,in accordance with various embodiments;

FIGS. 10A-10E illustrate an example process of fabricating an apparatus,in accordance with various embodiments;

FIGS. 11A-11C illustrate an example of fabricating an apparatus, inaccordance with various embodiments;

FIGS. 12A-12D illustrate various examples of fabricating an apparatus,in accordance with various embodiments;

FIGS. 13A-13C illustrate various examples of fabricating an apparatus,in accordance with various embodiments;

FIG. 14 illustrates an example device component with four layers ofgeometric structures, in accordance with various embodiments;

FIGS. 15A-15D illustrate examples of characterizing devices inaccordance with the present disclosure;

FIGS. 16A-16E illustrate example schematics of a supermode coupling in ametagrating, in accordance with various embodiments of the presentdisclosure;

FIGS. 17A-17D illustrate an example metagrating design using topologyoptimization, in accordance with various embodiments of the presentdisclosure;

FIGS. 18A-18C illustrate example supermode coupling, in accordance withvarious embodiments of the present disclosure;

FIGS. 19A-19F illustrate example experimental characterization ofmetasurfaces, in accordance with various embodiments of the presentdisclosure;

FIG. 20 illustrates an example mode analysis of a 75 degree beamdeflector designed using the optimization methodology, in accordancewith various embodiments of the present disclosure;

FIGS. 21A-21C illustrate an example coupled Bloch mode analysis of the75 degree metagrating for TE polarization, in accordance with variousembodiments of the present disclosure;

FIGS. 22A-22C illustrate an example coupled Bloch mode analysis of a75-degree deflector made of two TiO2 square pillars, in accordance withvarious embodiments of the present disclosure;

FIGS. 23A-23B illustrate examples of absolute deflection efficiency as afunction of different incident angles, in accordance with variousembodiments of the present disclosure;

FIGS. 24A-24B illustrate examples of tilted scanning electron microscopyimages of metagratings, in accordance with various embodiments of thepresent disclosure;

FIGS. 25A-25B illustrate example plots of experimental metagratingefficiencies as a function of incident wavelength of metagratings, inaccordance with various embodiments of the present disclosure;

FIGS. 26A-26B illustrate example plots of convergence of metagratings,in accordance with various embodiments of the present disclosure;

FIG. 27 illustrates an example top view of single period of a 75 degreemetagrating layout, in accordance with various embodiments of thepresent disclosure;

FIG. 28 illustrates the normalized and absolute efficiencies of a devicecomponent with one layer, two layers, and three layers of geometricstructures across a broadband spectrum, in accordance with variousembodiments;

FIG. 29 illustrates the normalized and absolute efficiencies of a devicecomponent with three layers and ten layers of geometric structuresacross a broadband spectrum, in accordance with various embodiments;

FIGS. 30A-30B illustrate example efficiency of a two layer devicecomponent as designed with two thickness, in accordance with variousembodiments;

FIGS. 31A-31B illustrate example efficiency of a three layer devicecomponent as designed with different thicknesses, in accordance withvarious embodiments;

FIGS. 32A-32B illustrate an example hyperspectral imaging platform thatincludes an apparatus comprising a plurality of device components andoptical properties of each device components of the apparatus across abroadband spectrum, in accordance with various embodiments;

FIGS. 33A-33C illustrate an example reconstructed spectra of variousincident waveforms as generated using the high resolution hyperspectralimaging device of FIG. 14A, in accordance with various embodiments;

FIG. 34 illustrates a plot of different phases of silicon, in accordancewith various embodiments;

FIGS. 35A-35D illustrate a theoretical analysis of the scatteringproperties of silicon ridges, in accordance with various embodiments;and

FIGS. 36A-36C illustrate the scattering spectra of individual siliconnanoridges, in accordance with various embodiments;

FIGS. 37A-37E illustrate example designs and performance of devices, inaccordance with various embodiments;

FIG. 38 illustrates an example of an optimization process, in accordancewith various embodiments; and

FIGS. 39A-39M illustrate various examples applications of an apparatus,in accordance with various embodiments.

While various embodiments discussed herein are amenable to modificationsand alternative forms, aspects thereof have been shown by way of examplein the drawings and will be described in detail. It should beunderstood, however, that the intention is not to limit the disclosureto the particular embodiments described. On the contrary, the intentionis to cover all modifications, equivalents, and alternatives fallingwithin the scope of the disclosure including aspects defined in theclaims. In addition, the term “example” as used throughout thisapplication is only by way of illustration, and not limitation.

DETAILED DESCRIPTION

Aspects of the present disclosure are believed to be applicable to avariety of different types of apparatuses, systems and methods involvinggeometrically optimized device components having optical properties fora particular optical response. In certain implementations, aspects ofthe present disclosure have been shown to be beneficial when used in thecontext of device components formed of at least one layer of silicongeometric structures and wherein portions of the device components arecombined together to form a periodic or aperiodic device and/orapparatus. In various specific aspects, the apparatus is used withand/or to form thin film solar cells, a hyper-spectral imaging system,various types of lenses such as a dielectric flat lens or a polarizationsensitive lens, a thermal management metasurface, a light emittingdevice, a fluorescence imaging system, a wearable flexible device,and/or a micro-electro-mechanical system (MEM), among other devicesand/or systems. Further, the device components can be formed of aplurality of layers of geometric structures. While not necessarily solimited, various aspects may be appreciated through a discussion ofexamples using such exemplary contexts.

Accordingly, in the following description various specific details areset forth to describe specific examples presented herein. It should beapparent to one skilled in the art, however, that one or more otherexamples and/or variations of these examples may be practiced withoutall the specific details given below. In other instances, well knownfeatures have not been described in detail so as not to obscure thedescription of the examples herein. For ease of illustration, the samereference numerals may be used in different diagrams to refer to thesame elements or additional instances of the same element.

In various instances, optics operate by bending light based on therefractive index of the material forming the optics. However, as devicesize is being reduced, such as with smartphones, the space that opticsoccupy is also limited and results in the optics being smaller than orcomparable to the wavelength of light. As previously discussed,metasurfaces are often used in optics. Metasurfaces are opticalhardware/devices that control a magnitude and phase response to lightbased on geometric layout of the metasurfaces. For example, ametasurface controls the wavefronts of incident electromagnetic wavesand support beam steering and focusing functionality with highefficiency. Metasurfaces, alternatively and/or in addition, specify thepolarization and angular momentum of light that exceeds the ability ofconventional optical components. Semiconducting geometric structures ascomponents in the metasurface result in high performance in the visiblewavelengths due to their high refractive indices and relatively lowabsorption losses. In accordance with various embodiments, geometricstructures of the metasurface are formed using silicon, such as crystalsilicon to provide broadband spectral responses, including the regime of400-500 nanometers (nm) which corresponds to blue wavelengths. Geometricstructures have a geometric shape and size defined by wavelength orsub-wavelength dimensions and having optical properties for a particularoptical response. Crystalline silicon, in various embodiments, possesseslow losses and is readily patterned into resonant geometric structures(e.g., nanostructures) within the 400-500 nm range and across abroadband spectrum range. A broadband spectrum range refers to orincludes a visible light, infrared and/or near-infrared range such as,for example, the broadband spectrum range 400-800 nm.

In various embodiments, the geometric structures are fabricated fromsilicon-on-insulator films bonded to a material, such as Pyrex. Forexample, the geometric structures are used to form beam deflectors thatsurprisingly operate across broadband ranges and are optimized forparticular wavelength ranges, such as for blue wavelengths. The silicongeometric structures, in various specific embodiments, are layered inmultiple layers. In some embodiments, using the geometric structures inat least one layer, as further described herein, a surface is designedthat is optimized for a variety of optical properties and/or responses.

In accordance with various embodiments, the geometric structures areused to form metasurfaces. For example, the geometric structures arewithin at least one layer of device components and can be stacked in atleast one dimension (x and/or z, and/or z, etc.). In other specificembodiments, the geometric structures are within multiple layers of thegeometric structures are stacked to form the device components inthree-dimensions (x, y, and z). The thickness of the device component(e.g., the direction of the layers, which may be the y-direction orz-direction depending on the point of view of the component) isdetermined by the thickness of each layer and the number of layers. Forinstance, the stacking of the layers forms the thickness of the devicecomponents. Each device component is geometrically shaped, via thegeometric structures, to have particular optical properties for anoptical response. For example, a device component is configured tocontrol an amplitude and phase of light for a particular wavelengthrange and/or across a broadband range. The device components arecombined to form an aperiodic or periodic device and/or apparatus. Forexample, each device component is a dielectric or semiconducting film orfilms and arranged on a flat substrate. In some specific embodiments,the device components are combined by stitching a portion of the devicecomponents in one or more directions, such as a width, length, and/orthickness/layer direction to form the periodic or aperiodic deviceand/or apparatus.

In these context, it is appreciated that “periodic” and “aperiodic”refer to structural aspects of the devices with which they areassociated. A periodic device and/or apparatus includes a metasurfaceformed of periodic structures, such as device components with geometricstructures of a periodic pattern (e.g., device components with regularlyarranged layers). As a more specific example, a periodic device and/orapparatus can be configured to modulate light in a periodic pattern. Anaperiodic device and/or apparatus includes a metasurface formed ofaperiodic structures, such as devices components formed of differentgeometric structures in an aperiodic pattern. Aperiodic devices and/orapparatuses, in various example embodiments, are formed of or using theplurality of device components and influence or control the amplitudeand phase of light across a broadband spectrum range.

Various specific embodiments are directed to optimization methods whichsupport compact optical hardware with capabilities that exceed those ofconventional diffractive optics. By tailoring optical hardware and/orsoftware, a foundation is created for energy efficient optical systemswith smartphone camera footprints, ranging from spectroscopic imagersand light field cameras to microscopes. An element of the hardware canbe the metasurface, which consists of or includes geometric structures(e.g., nanostructures) with optical properties specified by geometry.

There can be different methodology to design these devices. For example,an “optical nanomaterials database” can be created that consists of many(on an order of millions and billions) of different device components(e.g., nanoscopic metasurface building blocks) and/or geometricstructures therein. By efficiently searching through and combiningtogether different building blocks, followed by local refining, ablueprint is created for optically efficient metasurfaces. Themetasurfaces are tailored around energy efficient image processingfunctions or algorithms to optimize for system size and energyefficiency.

Relating to such optimization, various specific embodiments are directedto identifying: 1) the limits of amplitude and phase control in coupledgeometric structures (e.g., for example, these limits arepracticably-fundamental limits), and 2) the limits of coupling togetherdispersive (e.g., possessing wavelength-dependent optical responses)nanoresonators to yield broadband optical response (e.g., for example,these limits are physical or manufacturable-practicable limits). Incertain embodiments using this approach can be advantageous in that themetasurfaces are not necessarily constructed using a single layer ofsimple structures as found in conventional optical devices (sometimesreferred to as having physically-intuitive optical responses).

Various other specific embodiments are directed to methods foroptimizing a silicon-based metasurface. First, the metasurface layout isset (e.g., optimized) for a continuous profile (e.g., continuousdielectric constant), and then the continuous profile is converted intoa discrete profile (e.g., realistic dielectric constants). A layer ofthe metasurface can contain a unique layout of silicon-in-silicondioxide. These devices are created using an adjoint-based topologyoptimization process, and possess non-intuitive layouts that enable avariety of diffractive optics phenomena, such as beam deflection anddiffraction. In specific embodiments, the device components can betransmissive gratings that deflect a normally incident plane wave with aspecific polarization and wavelength. The initial device componentconsists of a random dielectric continuum of dielectric constants, withvalues ranging between the dielectric constants of the material formingthe geometric structures, such as of air and silicon. To improve theFigure of Merit (FoM), which corresponds to grating efficiency, aniterative process is performed that uses two electromagnetic simulationsper iteration, a forward and an adjoint simulation. These simulationsproduce two sets of electromagnetic field profiles within the device,which serve to simulate and/or specify specific changes in thedielectric constant at each location in a manner that improves the FoM.Over the course of multiple iterations, the dielectric continuum in thedevice converges to the dielectric constant of either silicon or air.The optimization method can be used to achieve multiple inputpolarizations and wavelengths, by performing forward and adjointsimulations for each optical degree of freedom per iteration.

The resulting device component(s) can form or be part of anapparatus/optical device that supports a plurality of optical modes. Theoptical modes are mediated by the bouncing of light between differentvertical interfaces of the layers within the device components. Forexample, light entering the device components can bounce within, insteadof going through the device components in a single path. When lightenters the device, the light itself can be considered an optical modeand each time the light (e.g., electromagnetic wave) bounces, a new modecan be generated. The different optical modes generated by the lightbouncing within the device can be generated by intra-mode couplings andinter-mode couplings. Intra-mode coupling refers to or includesreflection of an optical mode into itself (e.g., the optical modecouples with a reflection of itself into the device). Inter-modecoupling refers to or includes coupling of a first optical mode with asecond (or more) optical mode. Because of the bouncing of light withinthe device components, the devices support a plurality of spatiallyoverlapping optical modes per unit area. In specific embodiments,apparatuses formed can support at least three round trips of bouncingwithin device components and which recovers the steady state performanceof the device. The intra-mode coupling can be indicative of the numberof bounces.

The metasurface can include multiple layers of geometric structures, invarious embodiments. As a specific example, a 200 nm-thick fully visibletransmission grating is designed consisting of ten 20 nm-thick layers.At least two layers contain a unique layout of silicon-in-silicondioxide. The continuous and discrete profiles provide layouts withunexpected optical responses, and the efficiencies of these devices haveperformance metrics that, surprisingly, far exceed those of diffractiveoptics and single layer metasurface.

Various embodiments include a geometric structure and/or devicecomponent database that includes categorization of geometric structuresand/or starting points for topology of device components to achieveoptical properties for particular optical responses. In accordance withsome specific embodiments, optimization efforts can include: 1)generating the device component database and the optical responses ofthe geometrical structures, 2) combining together different devicecomponents (e.g., metasurface building blocks) in a way that provides aseamless material and optical interface, and 3) locally adjusting (e.g.,optimizing) and refining the device components to a definedspecification. In various embodiments, an optical nanomaterials databaseis generated, based on neutral network architecture and localoptimization function or algorithm, where a particular (e.g., desired)optical response is specified and the algorithm outputs thecorresponding metasurface.

Various embodiments disclosed herein include periodic or aperiodicdevices and/or apparatuses (e.g., metasurface devices) with tailoreddispersion to break the limit of classical optics (where images arefocused and recorded at an imaging plane, and processing images areinformationally inefficient and functionally limiting) and enable highperformance optical functionality in compact systems.

Turning now to the figures, FIG. 1A illustrates an example geometricstructure 102 in accordance with various embodiments. The geometricstructure 102 is a structure (e.g., nanostructure) formed of a materialand with optical properties specified by geometry. As illustrated, thegeometric structure 102 is layered on a base layer 100, as discussedfurther herein. The geometry, in various embodiments, includes ridges,cylinders, post, and spheres, among other shapes. However, the geometricshape is not limited to shapes common to metasurface engineering. Forexample, the shapes and/or layouts of the geometric structures includeshapes and layouts that provide such unexpected optical responses. Thespecific geometric structures are designed using a topology-optimizationmethodology, in which the resulting shapes or layouts of the structures(the specific geometry or shape which may not be pre-defined) support atleast three round trips of bouncing within device components and whichrecovers the steady state performance of the device.

In a number of embodiments, one or more of the geometric structuresinclude nanostructures. A nanostructure refers to or includes ageometric structure with one or more dimensions (e.g., length, width,and/or height/thickness) that are less than a micron (e.g., a magnitudeof a nanometer).

The geometric structure(s), in various embodiments, are formed of avariety of material to have and/or provide optical properties and/orresponses. For example, the geometric structures can be formed of one ormore dielectric or semiconducting material that are primarilynon-metallic and/or have a refractive index that is greater than two,such as insulating, semiconducting, phase change materials,electro-optic materials, and/or electrochemical materials. In variousspecific embodiments, the geometric structures are formed of silicon,such as polycrystalline silicon and/or crystalline silicon. As describedfurther herein, crystalline silicon has negligible absorption at greenand red wavelengths, and it is used to create efficient, fully visiblemetasurfaces. In various embodiments, each device component includes atleast one layer of geometric structures formed of two or more differentmaterials. The geometric structures can be formed by optimizing thetopology and boundaries, for each device component, to have particularoptical properties for a particular optical response and based onfabrication constraints. For example, the device component and/or theapparatus can have a particular optical response as a function of atleast one of: a number of layers of geometric structures of the devicecomponent, dimensions of the device component, thickness of each layerof the device component, materials forming the geometric structures,presence of a layer of solid material, and/or total thickness of thedevice component.

In various embodiments, the geometric structures are used to form amulti-layer device component. As further illustrated herein, themulti-layer device component is a stack of multiple layers of thegeometric structures. Each layer of geometric structures, for example,has a unique layout of one or more geometric structures. However,embodiments are not so limited, and can include device components havingone layer of geometric structures and/or having a single geometricstructure.

FIG. 1B illustrates an example of a device component 104 in accordancewith various embodiments. As illustrated, multiple layers 106-1, 106-2of geometric structures 102-1, 102-2, 102-3 (herein generally referredto as “the geometric structures 102”) provide a unique layout ofgeometric structures. In some specific embodiments, each layer 106-1,106-2 is a unique layout of geometric structures in silicon dioxide. Inother embodiments, one or more of the layers are formed of a solidmaterial, such a SiO2 spacer layer that is formed entirely of SiO2. Forexample, a layer formed of SiO2 may be between the two layers 106-1,106-2. As illustrated, the multiple layers 106-1, 106-2 of geometricstructures 102 are layered on a base layer 100, as discussed furtherherein. One or more of the layers 106-1, 106-2 is formed using two ormore materials. Example materials used to form the layers includesmetal, insulating, and/or semiconducting materials.

In accordance with various embodiments, the geometry of the devicecomponent 104 is optimized to have particular optical properties for aparticular optical response. For example, the layout of geometricstructures 102 of each layer is designed to provide optical propertiesfor the optical response. In various embodiments, the device component104 is configured to control the amplitude and phase of light in aparticular wavelength range and/or over a broad band. In some specificembodiments, the control includes a deflection angle, such as for adiffraction grating device. Other example optical properties includereflection, scattering, transmitting, etc.

In some specific embodiments, the device components are microstructures.A microstructure refers to or includes a structure with one or moredimensions that are a magnitude order of a micron. For example, in aparticular embodiment, a device component includes a one micron width(e.g., x-direction) and one micron length (e.g., z-direction), althoughembodiments are not so limited and the device components can be avariety of dimensions in an order of a micron magnitude. For example,the device components are microstructures formed of a plurality oflayers of nanostructures. Although the embodiment of FIG. 1B illustratestwo layers of geometric structures, embodiments in accordance with thepresent disclosure are not so limited and can include more or less thantwo layers.

In various embodiments, a plurality of device components are combined toform a device. The device formed is a surface (e.g., metasurface) withvarying geometries in a height/thickness dimension (e.g., y-direction).For example, the plurality of device components are combined bystitching portions of a plurality of device components together in twodirections (e.g., the x-direction and z-direction relative to they-direction/direction of the layers of the device components).

In some specific embodiments, the device components can be combined toform a transmission grating. A transmission grating can steer light to aspecific diffraction order based on the shapes of the geometricstructures of the device components. For particular deflection angles,gratings based on such concepts can steer light into a singlediffraction mode with particular efficiencies. Devices in this operationregime have large grating periods, and for metagratings, multiplewaveguide elements can be stitched within a single period tosufficiently sample a linear phase profile response. As the deflectionangle increases, the gratings become increasingly inefficient.

In accordance with various embodiments, a metagrating can be designedusing an adjoint-based topology optimization process. An adjoint-basedtopology optimization process can result in devices with a variety ofnon-intuitive layouts (as further illustrated by FIG. 16B among otherembodiments). As further described herein, the methodology can be usedfor designing transmissive gratings that deflect a normally incidentplane wave with a specific polarization and wavelength is summarized asfollows. The initial device consists of a random dielectric continuum ofdielectric constants, with values ranging between the dielectricconstants of the material forming the geometric structures, such as airand silicon. To improve the Figure of Merit (FoM), which corresponds tograting efficiency, an iterative process is performed that uses twoelectromagnetic simulations per iteration, a forward and an adjointsimulation. These simulations produce two sets of electromagnetic fieldprofiles within the device, which serve to simulate and/or specify(specific) changes in the dielectric constant at each location in amanner that improves the FoM. Over the course of multiple iterations,the dielectric continuum in the device converges to the dielectricconstant of either silicon or air. The boundaries between of the devicecomponents can be adjusted by accounting for fabrication constraintsduring the converging the continuous profile to the discrete profile.Surprisingly, the optimization method can extend to multiple inputpolarizations and wavelengths, by performing forward and adjointsimulations for each optical degree of freedom per iteration.

FIG. 1C illustrates an example device 110 in accordance with variousembodiments of the present disclosure. The illustration includes a topdown view (e.g., a birds-eye view) of the device. The device 110includes a plurality of device components 104-1, 104-2, 104-3, 104-4,104-5, 104-6, 104-7, 104-8, 104-N (herein generally referred to as “thedevice components 104”). As illustrated, a subset of the devicecomponents 104 are combined and/or stitched in two directions (e.g., thex-direction/width and the z-direction/length) to form the device 110. Invarious embodiments, the device is a metasurface, as described above andfurther described herein. The space between the device components 104can be set to mitigate or minimize coupling between adjacent devicecomponents 104 and/or to result in an approximate linear phase profileresponse.

Although the embodiments of FIG. 1C illustrates a device comprising ofthree rows and three columns of device components 104, embodiments inaccordance with the present disclosure are not so limited and caninclude a variety of number of device components combined to form thedevice and a variety of shapes. Further, the device components are notlimited to a square-like shape and can include other shapes, such asrectangular, circular, octagon, etc.

As previously described, the device supports a plurality of opticalmodes which can include inter-mode and intra-mode coupling that ismediated by the bouncing of light between different vertical interfacesof the layers of the device components. The different optical modesgenerated by the light bouncing within the device can be generated byintra-mode couplings and inter-mode couplings, as previously described.Because of the bouncing of light within the device components, thedevices support a plurality of spatially overlapping optical modes perunit area. In specific embodiments, apparatuses formed can support atleast three round trips of bouncing with device components and whichrecovers the steady state performance of the device.

FIGS. 2A-2C illustrate an example of an optimization process, inaccordance with various embodiments of the present disclosure. Aspreviously described, the optimization process can be referred to as anadjoint-based topology optimization. The adjoint-based topologyoptimization process can optimize a device (e.g., metasurface) using twooptimization processes: a topology optimization and a boundaryoptimization. For example, the topology optimization includes solvingMaxwell's equations and producing an output electromagnetic wave statefor a given input design and input electromagnetic wave condition. Theboundary optimization, in some instances, leverages a simulation engineto produce a fabricatible design (by imposing the fabrication toleranceinto the optimization) yielding an ideal output electromagnetic wavestate.

More specifically, the adjoint-based topology optimization process canuse two electromagnetic simulations per iteration to solve for gradientsin dielectric constants at each spatial location in the device. Theoptimization can be used for specific angles and multi-functionaldeflection, and can be used to solve for maximum transmission efficiencyof incident plane waves into targeted diffraction order channels.

FIG. 2A illustrates an example of simulated device, e.g., a grating 203.The grating 203 includes several geometric structures (e.g., asillustrated by particular geometric structure 212) formed of adielectric material (e.g., dielectric nanobeams). Light is incident onthe grating from the substrate 210 and couples into several diffractionorders. For example, forward and backward eigenmodes can be used forsimulation. The forward eigenmode represents the target transmitteddiffraction order in the forward simulations. The backward eigenmode canbe used for adjoint simulations. The forward and backward eigenmodes arethe same mode, but propagate in opposite directions. As illustrated byFIG. 2A, for a metagrating design for an incident plane wave with aspecific polarization and wavelength, the FoM can be defined as thetransmission intensity T_(m)to the target diffraction order m^(th):

FoM=T _(m),   EQ. 1

T_(m) can be obtained by projecting the total transmitted field to them^(th) diffraction order. For illustrative purposes and as illustratedby FIG. 2A, a plane wave can be considered that is normally incident ona one-dimensional lamellar grating through a substrate. The incidentpower P_(inc) in a single period of the grating is assumed to be 1. Thetransmission efficiency T_(m) can be normalized relative to theintensity of the incident beam, which can be found by performing anoverlap integral between two fields: the total field (E(x, z₀), H(x,z₀)) that is excited by the incident field, and the targetm^(th)diffraction order field (E_(m) ⁻(x, z₀), H_(m) ⁻(x, z₀)). In thisexpression for diffracted fields, the subscript “m” denotes the order ofthe diffraction channel and the subscript “−” denotes that the field ispropagation in downward in the −z direction. Both fields can beevaluated at the z=z₀ plane above the grating, and overlap integral isperformed for a grating period Λ:

$\begin{matrix}{{T = {{t}^{2} = \frac{{{\int_{x = x_{0}}^{x = {x_{0} + \Lambda}}{{\left\lbrack {{{E\left( {x,z_{0}} \right)} \times {H_{m}^{-}\left( {x,z_{0}} \right)}} - {{E_{m}^{-}\left( {x,z_{0}} \right)} \times {H\left( {x,z_{0}} \right)}}} \right\rbrack \cdot n_{2}}{dx}}}}^{2}}{{N_{m}}^{2}}}},} & {{EQ}.\mspace{14mu} 2}\end{matrix}$

Where N_(m)=∫_(x=x) ₀ ^(x=x) ⁰ ^(+Λ)[E_(m) ⁺(x, z₀)×H_(m) ⁻(x, z₀)−E_(m)⁻(x, z₀)×H_(m) ⁻(x, z₀)]·n₂dx defines the normalization of the m^(th)diffraction order, and (E_(m) ⁺(x, z₀), H_(m) ⁺(x, z₀)) and (E_(m) ⁻(x,z₀), H_(m) ⁻(x, z₀)) denotes the fields propagating in the forward (+z)and backward (−z) directions. This normalization procedure is consistentwith the Lorentz reciprocity relation, and can be valid for both lossy(e.g., plasmonic) and lossless (e.g., fiber modes or planewaves)eigenmodes. Assuming |N_(m)|=1 the FoM can be written as:

FoM=|∫_(x=x) ₀ ^(x=x) ⁰ ^(+Λ)[E(x, z₀)×H_(m) ⁻(x, z₀)−E_(m) ⁻(x,z₀)×H(x, z₀)]·n₂dx|²   EQ. 3

If a perturbation is introduced in permittivity Δε within a volume ΔV ata location r₁=(x₁, z₁) in the grating layer, the total field at the z=z₀plane can be defined at [E(x, z₀)+ΔE(x, z₀), H(x, z₀)+ΔH(x, z₀)] and thechanged FoM′ can be defined as:

$\begin{matrix}{{FoM}^{\prime} = {{\int_{x = x_{0}}^{x = {x_{0} + \Lambda}}{{\begin{bmatrix}{{\left( {{E\left( {x,z_{0}} \right)} + {\Delta \; {E\left( {x,z_{0}} \right)}}} \right) \times {H_{m}^{-}\left( {x,z_{0}} \right)}} -} \\{{E_{m}^{-}\left( {x,z_{0}} \right)} \times \left( {{H\left( {x,z_{0}} \right)} + {\Delta \; {H\left( {x,z_{0}} \right)}}} \right)}\end{bmatrix} \cdot n_{2}}{dx}}}}^{2}} & {{EQ}.\mspace{14mu} 4}\end{matrix}$

The expression can be simplified by omitting the 0(Δ²) terms, which canbe a valid approximation if the change in the field from the dielectricperturbation Δε is less than a threshold amount. The change of the FoMresulting from the inclusion of Δε can be found by combining EQ. 3 andEQ. 4:

ΔFoM=FoM′−FoM=2Re(conj(t)∫_(x=x) ₀ ^(x=x) ⁰ ^(−Λ)[E(x, z₀)×H_(m) ⁻(x,z₀)−E_(m) ⁻(x, z₀×H(x, z₀)]·n₂dx)   EQ. 5

The addition of the perturbation Δε at location r₁ can be treated as theinsertion of an electric dipole with dipole moment p=ε₀ΔεΔE_(app), whereE_(app) denotes the approximate value of the electric field inside theperturbation. An estimate of E_(app) can include E_(app)≈E(r₁). Thiselectric dipole can produce scattered fields described by:

$\begin{matrix}\left\{ \begin{matrix}{{\Delta \; {E\left( {x,z_{0}} \right)}} = {\omega^{2}{G_{ep}\left( {r,r_{1}} \right)}p}} \\{{\Delta \; {H\left( {x,z_{0}} \right)}} = {\omega^{2}{G_{h\; p}\left( {r,r_{1}} \right)}p}}\end{matrix} \right. & {{EQ}.\mspace{14mu} 6}\end{matrix}$

Where G_(ep) and G_(hp) are the Green's tensors. EQ. 5 can be rewrittenas:

ΔFoM=2ω²ε₀Δε(r₁)ΔVRe (conj(t)∫_(x=x) ₀ ^(x=x) ^(x) ₀^(+Λ)[(G_(ep)(r,r₁)E(r₁))×H_(m) ⁻(x,z₀)−E_(m)⁻(x,z₀)×(G_(hp)(r,r₁)E(r₁))]·n₂dx)   EQ.7

Applying vector identities into EQ. 7, the gradient of FoM to the localpermittivity at r₁=(x₁, z₁) can be defined as:

$\begin{matrix}{{\frac{\partial{FoM}}{\partial ɛ}}_{r = r_{1}} = {2ɛ_{0}\omega^{2}\Delta \; {{VRe}\left( {{{conj}(j)}{{E\left( r_{1} \right)} \cdot {\int_{x = x_{0}}^{x = {x_{0} + \Lambda}}{\left\lbrack {\left( {{{G_{ep}\left( {r_{1},r} \right)}\left( {{- n_{z}} \times {H_{m}^{-}\left( {x,z_{0}} \right)}} \right)} + {{- {G_{h\; p}^{T}\left( {r_{1},r} \right)}}\left( {n_{z} \times {E_{m}^{-}\left( {x,z_{0}} \right)}} \right)}} \right\rbrack {dx}} \right).}}}} \right.}}} & {{EQ}.\mspace{14mu} 8}\end{matrix}$

The term [G_(ep)(r₁, r)(−n_(x)×H_(m) ⁻(x, z₀))+(−1)G_(hp)^(T)(r₁,r)(n_(z)×E_(m) ⁻(x,z₀))] represents the field induced by anincident field (E_(m) at the location r₁. EQ. 8 can be rewritten as:

$\begin{matrix}{{\frac{\partial{FoM}}{\partial ɛ}}_{r = r_{1}} = {2ɛ_{0}\omega^{2}\Delta \; {{{VRe}\left( {{{conj}(t)}{{E\left( r_{1} \right)} \cdot {E_{adjoint}\left( r_{1} \right)}}} \right)}.}}} & {{EQ}.\mspace{14mu} 9}\end{matrix}$

Where E_(adjoint) (r₁)=[G_(ep)(r₁, r)(−n_(z)×H_(m) ⁻(x,z₀))+(−1)G_(hp)^(T)(r₁, r)(n_(z)×E_(m) ^(−(x,z) ₀))] is defined as the adjoint fieldand can be obtained by an auxiliary simulation in which thebackward-propagating eigenmode (E_(m) ⁻, H_(m) ⁻) serves as the incidentfield. EQ. 9 indicates that with two simulations, one forward simulationand one adjoint simulation,

$\frac{\partial{FoM}}{\partial ɛ}$

can be evaluation at each (e.g., all) locations of the device.

FIG. 2B illustrates an optimization process. As illustrated, theoptimization process can be performed in an iterative manner, which inspecific experimental embodiments can include 200-300 simulatediterations to achieve convergence. The starting point 214 can be arandom dielectric continuum of dielectric constants, and over the courseof the iterative process, this continuum can converge to discrete valuesof silicon and air. A base pattern p can be obtained from the startingpoint, at 216. A blurring function is applied to the base pattern,resulting in a blurred geometry {tilde over (p)}=B (p) that is splitinto three geometric versions: an eroded pattern 220(p^(−e)=Q^(e)({tilde over (p)})), an intermediate pattern 222 (e.g.,p^(−i)=i({tilde over (p)})), and the dilated pattern 224(p^(−d)=Q^(d)({tilde over (p)})), at 218. Over each iteration, forwardand adjoint simulations are performed for the eroded, intermediate, anddilated patterns at 226 which produces values of G 228, 230, 232 (e.g.,G ^(e), G ^(i), G ^(d)) that correspond to the gradient of the figure ofmerit, ∂(FoM)/∂ε. These values can then be used to modify the dielectricconstant at all points of the device. For example, the produced values Gare combined to generate a total gradient value G_(tot), at 234. Thetotal gradient value can be defined as

$G_{tot} = {\sum\limits_{q}^{\;}{{\overset{\_}{G}}^{q}{\frac{{\partial\rho^{- q}}{\partial\overset{\sim}{p}}}{{\partial\overset{\sim}{p}}{\partial\rho}}.}}}$

A bias function is added to the total gradient value to push the patterntoward binary structures (e.g., binary push), at 236. The binary pushcan be defined by G=G_(tot)+b(2p−1)^(2n). And, gradient and constraintsare applied, at 238 and 240. The gradient constraint can be defined asp_(new)=p+cG. For example, the constraints can be applied periodically(e.g., every tens of iteration) to remove geometric structures that aresmaller than a threshold size. The process is repeated until convergenceoccurs at 242.

During the topology optimization, the dielectric constant at each pointin the device can range in a continuous fashion between ε_(high)=ε_(si)and ε_(low)=ε_(air). For a given iteration u, the device can be encodedat all spatial locations r_(i) with a real number p_(u)(r_(i))∈[0,1],such that the dielectric constant at each point is given by∈_(u)(r_(i))=p_(u)(r_(i))∈_(high)+(1−pu(r_(i)))∈_(low).

In order to mitigate (e.g., avoid or eliminate) features of less than athreshold size and create devices that are robust to fabrication error,a robustness algorithm can be applied. For more general and specificinformation related to a robustness algorithm, reference is made to F.Wang, et al, “Robust Topology Optimization of Photonic CrystalWaveguides with Tailored Dispersion Properties,” J. Opt. Soc. Am. B, 28,387-397 (2011), which is hereby incorporated by reference for itsteaching. At the beginning of iteration u, a blurring function B_(u) isapplied to the device p_(u), B_(u):R^(N)→R^(N), were N is the totalnumber of points in the device:

$\begin{matrix}{{{B_{u}\left( {p_{u}\left( r_{i} \right)} \right)} = {\frac{\sum\limits_{j \in N_{e}}^{\;}\left\lbrack {\left( {R_{u} - {{r_{j} - r_{i}}}} \right){p_{u}\left( r_{j} \right)}} \right\rbrack}{\sum\limits_{j \in N_{e}}^{\;}\left\lbrack {\left( {R_{u} - {{r_{j} - r_{i}}}} \right){p_{u}\left( r_{j} \right)}} \right\rbrack} = {{\overset{\sim}{p}}_{u}\left( r_{i} \right)}}},{N_{e} = \left\{ j \middle| {{{r_{j} - r_{i}}} \leq R_{u}} \right\}}} & {{EQ}.\mspace{14mu} 10}\end{matrix}$

R_(u) is the blurring radius and can be iteration-dependent. Thisblurred geometry is then split into three separate geometric versionsp_(u) ^(−q)(r_(i)), where q∈{E,I,D} corresponds to the eroded,intermediate, and dilated devices.

FIG. 2C illustrates examples of a base pattern 243, an eroded pattern245, an intermediate pattern 247, and a dilated pattern 249, inaccordance with varies embodiments. The patterns can be designed, inspecific experimental embodiments, for a 75 degree metagrating (e.g., asillustrated by the experimental embodiments further described herein).The green curve represents the boundary of the base pattern 243. Theintermediate pattern 247 represents the ideal pattern for fabrication,and the eroded and dilated patterns 245, 249 represent devices that areover-etched and under-etched, respectively, during fabrication. Thesegeometric variants of the device can be mathematically described as:

$\begin{matrix}{{p_{u}^{- q}\left( r_{i} \right)} = \begin{Bmatrix}{n_{u}^{q}\left\{ {{\exp\left\lbrack {{- {\beta_{u}\left( {1 - {{{\overset{\sim}{p}}_{u}\left( r_{i} \right)}/n_{u}^{q}}} \right\rbrack}} - {\left( {1 - \frac{{\overset{\sim}{p}}_{u}\left( r_{i} \right)}{n_{u}^{q}}} \right){\exp \left( {- \beta_{u}} \right)}}} \right\}},} \right.} \\{0 \leq {{\overset{\sim}{p}}_{u}\left( r_{i} \right)} \leq n_{u}^{q}} \\{{\left( {1 - n_{u}^{q}} \right)\left\{ {1 - {\exp \;\left\lbrack \frac{- {\beta_{u}\left( {{{\overset{\sim}{p}}_{u}\left( r_{i} \right)} - n_{u}^{q}} \right)}}{\left( {1 - n_{u}^{q}} \right)} \right\rbrack} + \frac{\left( {{{\overset{\sim}{p}}_{u}\left( r_{i} \right)} - n_{u}^{q}} \right)}{\left( {1 - n_{u}^{q}} \right)e^{- B_{u}}}} \right\}} +} \\{n_{u}^{q},{n_{u}^{q} \leq {{\overset{\sim}{p}}_{u}\left( r_{i} \right)} \leq 1}}\end{Bmatrix}} & {{EQ}.\mspace{14mu} 11}\end{matrix}$

Where B_(u) the iteration-dependent sharpness of the threshold function,and n_(u) ^(q) is the cutoff value of {tilde over (p)}_(u) at which thethreshold is applied, such that n_(u) ^(E)>0.5, n_(u) ^(I)=0.5, andn_(u) ^(D)<0.5.

Once the forward and adjoint calculations are performed for all threegeometric variants of the device, G ^(q)(r_(i))=∂FoM^(q)/∂∈ can becalculated for each geometry. A singular ∂(FoM)/∂∈ can be applied to thebase pattern and that combines the three G ^(q) can be expressed as:

$\begin{matrix}{{G_{tot}\left( r_{i} \right)} = {\sum\limits_{q}^{\;}{\sum\limits_{e \in N_{1}}^{\;}{{{\overset{\_}{G}}^{q}\left( r_{e} \right)}\frac{{\partial{{\overset{\_}{p}}_{u}^{q}\left( r_{e} \right)}}{\partial{{\overset{\sim}{p}}_{u}\left( r_{e} \right)}}}{{\partial{{\overset{\sim}{p}}_{u}\left( r_{e} \right)}}{\partial{p\left( r_{i} \right)}}}}}}} & {{EQ}.\mspace{14mu} 12}\end{matrix}$

Once this gradient is calculated, a bias function Y_(u)(p_(u)(r_(i))) isadded, which can be used to push the pattern towards a binary structure(e.g., p_(u)(r_(i))=[0,1]). This bias function increases in strength asthe full iterative optimization process nears completion. In the case ofa multi-functional device with M functions, each withG_(m)(r_(i))=∂FoM_(m)/∂∈, different ways to combine G_(m) into a singleG can be used. Two such methods include:

G(r _(i))=ΣmG _(m)(r _(i)), and

G(r _(i))=ΣmG _(m)(r _(i))(T _(m) ^(tgt) −T _(m))   EQ. 13

The topology method optimizes for the total combined power going intoall targeted diffraction channels. The bottom method minimizes the meansquared error for a set of targeted power transmission coefficientsassociated with each diffraction channel, T_(m) ^(tgt).

In order to remove (tiny) nanoscale features in the device design, acircular spatial blurring filter to the material distribution (r_(i))periodically, such as every few tens of simulation iterations. In aspecific experimental embodiment, a diameter of this circular blurringfilter is approximately 80 nm, and filtering is performed every 40iterations. This filtering results in reductions in efficiency, whichcan be visualized as sharp dips, such as further described andillustrated herein in connection with FIG. 17D of the experimentalembodiments.

FIGS. 3A-3C illustrate an example of a continuous profile and a discreteprofile of a topology of a device component in accordance with variousembodiments. In various embodiments, a device component is configured todeflect a particular wavelength range, such as blue wavelengths (400-500nm) at an optimized efficiency. The device component can include asingle layer of geometric structures or multiple layers of geometricstructures.

In some experimental/representative embodiments, the device componentincludes a 200 nm-thick (e.g., layered direction) fully visibletransmission grating consisting of ten 20 nm-thick layers of geometricstructures. Each of the ten layers includes a unique layout of siliconin silicon dioxide. In other experimental/representative embodiments, asubset of the layers include unique layouts of geometric structures andthe remaining one or more layers include solid material (e.g., spacers)as previously discussed. A layer of solid material, in some instances,is used as a spacer between two layers of geometric structures.Optimized, in various embodiments, is not limited to absolute optimalvalue but rather is an efficiency for the particular optical propertyand/or response that is above a predefined threshold. In some specificembodiments, the predefined threshold include 60 percent, although otherembodiments are not so limited and can include thresholds between 60-80percent or more. The embodiment type of FIGS. 3A-3C is a blazed grating,whereas embodiments in accordance with the present disclosure includeother types of diffraction gratings and optical devices.

One way that device components can be designed is by setting (e.g.,optimizing) a topology of the layer(s) of geometric structures. Forexample, a continuous profile is first set (e.g., optimize) to haveparticular optical property for particular optical response. Then, thecontinuous profile is converted to a discrete profile (e.g., binary).FIG. 3A, for example, illustrates a cross-section of a continuousprofile configured to deflect blue wavelengths at an optimizedefficiency. The continuous profile includes a range of material fromsilicon to air, with silicon being light, oxide being dark, and valuesbetween including a mixture of the silicon and air. FIG. 3B illustratesa cross section of a discrete profile as generated by converting thecontinuous profile illustrated by FIG. 3A to a discrete result. Forexample, the resulting discrete profile includes silicon (light) andoxide (dark).

In various embodiments, multiple layers of geometric structures can beused. FIG. 3C illustrates the efficiencies of the device components ofFIG. 3A and FIG. 3B having multiple layers of geometric structures andwith metrics that exceed those of diffractive optics and single layersof geometric structures. In various embodiments, the discrete profileincludes structures that are binary and that are generated withoutcross-sections.

In accordance with various embodiments, the device components areoptimized using topology optimization, boundary optimization, or both.Topology optimization sets the number and shape of elementary geometries(e.g., geometric structures) within a device layout, and boundaryoptimization adjusts and/or refines the boundaries of the geometriesgenerated in the topology optimization process. For example, the edgesbetween boundaries of the device components can be periodically adjustedby accounting for fabrication constraints during the converging thecontinuous profile to the discrete profile. In some embodiments, thetopology optimization process effectively generates geometries that arenear the theoretical global optimum with high probability. Thesegeometries are then used as starting points in the boundary optimizationroutine, which further refines the geometry as well as imposesconstraints on minimum feature size for fabrication purposes.

FIGS. 4A-4D illustrate an example of boundary optimization of devicecomponents containing two geometric structures and three geometricstructures (in a layer or more). The device component, as illustrated,is a flat blazed grating that is configured to deflect light at a 45degree angle. As previously discussed, the term optimized as used hereinis not limited to the best possible outcome but rather to an optimalvalue at or above an acceptable threshold. For example, with optics,there can be many local maximums when optimizing for a particularoptical property and response. In various embodiments, a startingtopology for the device components is set and then adjusted forboundaries. As illustrated by FIG. 4A, a first device component includestwo geometric structures 402-1, 402-2 and as illustrated by FIG. 4C asecond device component includes three geometric structures 402-3,402-4, 402-5. FIG. 4B and FIG. 4D illustrate boundary optimizationresults from the first and second device components, and specificallyillustrate the probability of obtaining device component with aparticular efficiency after 1,000 realizations using boundaryoptimization from a starting point. For example, as illustrated by FIG.4B, the boundary optimization of the first device component results inan efficiency of seventy-percent about fifteen-percent of the time. FIG.4D illustrates that boundary optimization of the second device componentresults in an optical property (efficiency) of seventy-percent aboutfive-percent of the time.

FIG.5 illustrates an example of topology and boundary optimization of adevice component with two geometric structures. The device component, invarious embodiments, includes the first device component as illustratedby FIG. 4A and that is configured to deflect light at a 45 degree angle.As illustrated, using both the topology and boundary optimizationresults in an optical property of seventy-percent about fifty-percent ofthe time. That is, there is around a fifty percent probability ofobtaining a device component with an optical property of seventy percentafter 1,000 realizations when performing both topology and boundaryoptimization.

FIGS. 6A-6B illustrate an example boundary optimization of a devicecomponent that is optimized to deflect light at a 20 degree angle. Forexample, FIG. 6A illustrates a device component with five geometricstructures configured to deflect light at 20 degree angle. Asillustrated, compared to the 45 degree angle, the device componentincludes a greater number of geometric structures. FIG. 6B illustratesthe probability of optimizing a device to a particular efficiency afterten thousand realizations using boundary optimization. As illustrated,there is a 0.01 probability of obtaining a device with a seventy percentefficiency after ten-thousand realizations.

FIGS. 7A-7D illustrate an example optimization of a device componentthat is configured to deflect light at a 20 degree angle. For example,FIG. 7A illustrates a device component with five geometric structuresoptimized to deflect light at 20 degree angle. FIG. 7B illustrates theprobability of optimizing a device component to a particular efficiencyafter ten thousand realizations using only boundary optimization. Asillustrated, there is a 0.01 probability of obtaining a device with aseventy-percent efficiency. FIG. 7C illustrates the probability ofoptimizing a device component to a particular efficiency using topologyoptimization and boundary optimization from a starting point. FIG. 7Dillustrates the probability of optimizing a device component to aparticular efficiency using topology optimization and boundaryoptimization from a constant profile (with low average dielectric).

FIGS. 8A-8D illustrate example device components with geometricstructures, such as the geometric structure illustrated by FIG. 1A,optimized for broadband spectrum transmission efficiency. For example,FIGS. 8A-8C illustrate a cross-section of a grating with geometricstructures that are locally optimized for transmitting differentwavelengths of light (e.g. 450 nm, 500 nm, and 550 nm). FIG. 8Dillustrates the relative efficiency and absolute efficiency across abroadband spectrum. As illustrated, the particular geometric structure,which includes an “L-shape” has a near one-hundred percent relativeefficiency at around 450 nm.

FIGS. 9A-9B illustrate example device components optimized for broadbandspectrum reflection efficiency. FIG. 9A illustrates a cross-section of agrating with geometric structures that are locally optimized forreflecting different wavelengths of light (e.g. 430 nm, 470 nm, 510 nm,550 nm, 590 nm, and 630 nm). The geometric structures, in variousembodiments, include the geometric structure illustrated by FIG. 1A.FIG. 9B illustrates the relative efficiency and absolute efficiencyacross a broadband spectrum. As illustrated, the particular geometricstructure, which includes an “L-shape” has a near one-hundred percentrelative efficiency at around 560 nm. FIG. 9C illustrates the relativetransmission efficiency and absolute transmission efficiency of light oftwo polarization states (transverse electric (TE) and transversemagnetic (TM)) for the polarization-insensitive device componentsillustrated by FIG. 1A.

FIG. 10A-10E illustrate an example process of fabricating an apparatus,in accordance with various embodiments. As previously discussed,apparatuses in accordance with the present disclosure include at leastone planar layer of geometric structures. In specific embodiments,multiple planar layers of geometric structures can be stacked one on topof the other, to yield a three-dimensional optical apparatus. Each layerincludes two or more distinct materials that are patterned, such thatthe materials cumulatively fill the full layer area. Further, the deviceinclude device components that have a constant cross-section extendingalong one axis (e.g., for example, an ordered stack of logs), andmulti-layer device components that are three-dimensional and lack atranslational symmetry constant.

A variety of material is used for a substrate for the device components,including rigid semiconducting or insulating substrates. These layerscan also be directly fabricated onto and integrated with structuresincluding but not limited to CCD imaging arrays, silicon photonic ICs,photodetectors, solar cells, LEDs, lasers, quantum cascade lasers, andtransparent displays. Further, the layers can be transfer-printed ontoalternative substrates, such as polymers, plastics, and siliconeelastomers, using wafer bonding or pick-up and transfer techniques.Integration of geometric structures with flexible and stretchablesubstrates is made possible due to the intrinsic flexible mechanics.Further, the geometric structures can processed directly into opticalMEMS structures, and their thin film form factor enables high frequencyand high mechanical quality factor operation.

Each layer is defined using a combination of at least two generalprocesses. In the first, geometric structures comprising distinctmaterials are defined using additive or subtractive manufacturing. Inthe second, the layer is planarized, such that its top surface ispolished down or an additional material is added to produce a flatsurface at the top of the layer. These two processes can be repeated foran individual layer to produce layers that consist of more than twomaterials. As used herein, when referring to “layer 1, layer 2, etc.”,layer 1 refers to the layer that is processed first and therefore deeper(e.g., closer to) within the substrate, and higher numbers are processedsubsequently and therefore closer to the surface of the substrate. Whenreferring to “material 1, material 2, etc.”, these refer to the order inwhich materials are deposited into each individual layer. Thesematerials may differ in different layers, in a number of embodiments.Geometric structures within individual layers can be defined using abroad range of additive and subtractive manufacturing techniques. FIGS.10A-10E illustrate an example of a series of general steps that can beused to define the geometric structures in a layer using an additiveprocess. Although FIGS. 10A-10E illustrate a single layer, the processwhich is illustrated is repeated, in various embodiments, for aplurality of layers of geometric structures, although some embodimentsinclude a single layer of geometric structures.

As illustrated by FIG. 10A, the first layer includes a base layer 1060.The base layer 1060, in various embodiments, includes a thin filmmaterial layer and/or the top layer of a substrate. In accordance withsome embodiments, the substrate is formed of a rigid semiconducting orinsulating material. As a specific example, the substrate is formed ofglass. Alignment marks 1062 are added to the base layer 1060 and usedthroughout the patterning process. For example, the alignment markers1062 are used to pattern each layer to ensure that the layers arealigned with respect to one another with precision.

A sacrificial material 1064 is added to the second layer, as illustratedby FIG. 10B. The sacrificial material 1064, in various embodiments,includes a photoresist material. The material 1064 is deposited onto thebase layer 1060, as illustrated by FIG. 10B, and then patterned, asillustrated by FIG. 10C using one or more of a number of processes. Forexample, the processes includes optical lithography, electron beamlithography, direct write lithography, nanoimprint lithography, and/orchemical self-assembly. These patterns are aligned relative to the otherlayers using the alignment marks 1062.

Further, as illustrated by FIG. 10D, another material 1066 is depositedon a first layer (e.g., onto a surface of the base layer 1060). Forexample, the other material 2166 is deposited using a technique such asatmospheric pressure chemical vapor deposition (CVD), low-pressure CVD,ultrahigh vacuum CVD, aerosol assisted CVD, direct liquid injection CVD,microwave plasma-assisted CVD, plasma-enhanced CVD, remoteplasma-enhanced CVD, atomic layer CVD (also known as ALD), combustionCVD, hot filament CVD, hybrid physical-chemical vapor deposition,metalorganic CVD, rapid thermal CVD, photo-initiated CVD, sputtering,electron beam evaporation, thermal evaporation, wet chemical processing,or ion beam deposition. And, as illustrated by FIG. 10E, the sacrificialmaterial 1064 is removed, which leaves a negative pattern of thematerial 1066 on the base layer 1060 (e.g., the substrate).

Alternatively and/or in addition, the geometric structures arefabricating using a different additive process. FIGS. 11A-11C illustrateanother such fabrication process.

FIGS. 11A-11C illustrate an example of fabricating an apparatus, inaccordance with various embodiments. As illustrated by FIG. 11A, thefirst layer includes a base layer 1160 with alignment markers 1162, suchas those described above by FIGS. 10A-10E. The material 1166, which caninclude the same material 1166 as illustrated by FIGS. 11C-11E, isdeposited onto the base layer 1160, as illustrated by FIG. 11B to formthe first layer. In various embodiments, the material 1166 is depositedonto the base layer 1160 as previously described in connection with FIG.10 and/or attached using wafer bonding. Examples of wafer bondinginclude direct, surface activated, plasma activated, anodic, eutectic,glass frit, adhesive, thermocompression, reactive, and transient liquidphase diffusion bonding. Next, the material 1166 is patterned and thenetched using a wet or dry etching technique. The etching, in someembodiments, is completely or partially through the material film. Theresult is a geometric structure formed of the material 1166 within thefirst layer, as illustrated by FIG. 11C.

FIGS. 12A-12D illustrate various examples of fabricating an apparatus,in accordance with various embodiments. In a number of embodiments,geometric structures of one or more layers are formed of more than onematerial. In such embodiments, the geometric structures are patternedusing a process that combines material deposition onto a materialconsisting of geometric structures, followed by chemical-mechanicalpolishing (CMP). For example, FIG. 12A illustrates a base layer 1260with a geometric structure formed of a first material 1266. The devicecomponent illustrated by FIG. 12A is formed via the process illustratedand described by FIGS. 11A-11C or FIGS. 10A-10E in various embodiments.

The regions of the first layer (e.g., the layer on the base layer 1260)that are not filled with the first material contain air. As illustratedby FIG. 12B, a second material is deposited and the layer is CMPpolished, leaving geometric structures comprising the second material1268-1, 1268-2.

In various embodiments, one or more layers consists of more than twomaterials (e.g., the geometric structures comprising the first material1266 and the second material 1268-1, 1268-2). In a number of embodimentsto create a first layer with three materials, the process described byFIGS. 12A-12D is used. For example, the first layer defined by firstdepositing and patterning a first material 2366 (i.e. transparent oxidelayer), followed by the deposition and CMP polishing of a secondmaterial 1268-1, 1268-2 as illustrated by FIGS. 12A-12B. The layer ispatterned again, followed by the deposition of CMP polishing of a thirdmaterial 1270 as illustrated by FIG. 12C and FIG. 12D. In otherembodiments and/or in addition, as further illustrated herein, the thirdmaterial includes a planarization material.

FIG. 13A-13C illustrate various examples of fabricating an apparatus, inaccordance with various embodiments. In various embodiments, one or moreof the layers of geometric structures is planarized. Planarization isused to ensure that the thickness of each layer is well-defined and tosteamline the manufacturing of the device component. Planarization of anindividual layers, in some instances, is achieved by spin-coating aplanarization material onto the substrate, followed by material curing.The planarization material includes a liquid, such as spin-on-glassesand epoxy resins. This method ensures that the top surface of the layeris smooth and planar. The total thickness of the layer is controlled bycontrolling the spin speed during spin coating. Alternatively and/or inaddition, an individual layer is planarized using CMP. For example, FIG.13A illustrates planarization of a layer comprising one geometricstructure of a first material 1366 using a planarization material 1376(and a base layer 1360). FIG. 13B illustrates planarization of a layercomprising one geometric structure of a first material 1366 and twogeometric structures of a second material 1368-1, 1368-2 using aplanarization material 1376. FIG. 13C illustrates planarization of alayer comprising one geometric structure of a first material 1366, twogeometric structures of a second material 1368-1, 1368-2, and onegeometric structure of a third material 1370 using a planarizationmaterial 1376.

Various embodiments include a combination of these processes of defininggeometric structures and planarization to yield layers consisting of twoor more materials. For example, to create a layer with three materials,an additive process to define geometric structures is performed twice todefine nanostructures consisting of material one and material two,followed by planarizing the layer using material three. In someembodiment, the curable planarization liquid is the third material.

A broad range of materials are incorporated into the fabrication ofapparatuses in accordance with the present application and depending onthe particular application. These include metals, semiconductors, andinsulators; linear and non-linear optical materials; active and passiveelectronic materials. Different material combinations are used to targetapplications at specific wavelengths or wavelength ranges. In someembodiments, wavelength ranges are defined as follows:

-   -   Visible: 400 nm-800 nm wavelengths;    -   Near-infrared (NIR): 800 nm-2000 nm wavelengths;    -   Mid-infrared (MIR): 2 μm-10 μm wavelengths; and    -   Far-infrared (FIR): 10 μm-40 μm wavelengths.        Further in various specific examples, material is selected based        on:    -   Si (crystalline): High index (>3.4), visible and NIR        applications    -   Si (polycrystalline): High index (>3.4), red and NIR        applications    -   Si (amorphous): High index (>3.4), NIR applications    -   SiO2: index ˜1.5, visible and NIR applications    -   Al2O3: index ˜1.7, UV, visible and NIR applications    -   GeO2: index ˜1.7, UV, visible and NIR applications    -   MgO: index ˜1.7, NIR and MIR applications    -   TeO2: index ˜2.2, MIR applications    -   TiO2: index ˜2.5, visible and NIR applications    -   HfO2: index ˜2.1, visible and NIR applications    -   ZrO2: index ˜2.1, UV, visible and NIR applications    -   AlAs: index ˜3, NIR applications    -   GaAs: index ˜3.6, NIR applications    -   InAs: index ˜3.4, MIR applications    -   KBr: index ˜1.5, UV, visible, NIR, MIR, and FIR applications    -   C (diamond): index ˜2.4, UV, visible, NIR, and MIR applications    -   CaCO3: index 1.4-1.9, birefringent, UV, visible, and NIR        applications    -   SiC: index 3-3.5, NIR and FIR applications    -   Photonic material (e.g., SiC): MIR applications    -   AgCl: index ˜2, NIR and MIR applications    -   CsCl: index ˜1.6, visible, NIR, and MIR applications    -   KC1: index ˜1.4, UV, visible, NIR, MIR, and FIR applications    -   NaCl: index ˜1.5, visible, NIR, MIR, and FIR applications    -   BaF2: index ˜1.45, UV, visible, NIR, and MIR applications    -   CaF2: index ˜1.4, UV, visible, NIR, and MIR applications    -   MgF2: index ˜1.35, UV, visible, NIR, and MIR applications    -   Ge: High index (-4), NIR, MIR, and FIR applications    -   AIN: index ˜2, UV, visible, NIR, and MIR applications    -   Si3N4: index ˜2, visible and NIR applications    -   LiNbO3: index ˜2.2, visible and NIR applications    -   GaP: High index (>3), visible and NIR applications    -   As2S3: index ˜2.4, NIR and MIR applications    -   CS2: index ˜1.6, visible and NIR applications, highly dispersive    -   ZnS: index ˜2.2, visible, NIR, and MIR applications    -   AlSb: High index (˜3.3), NIR applications    -   ZnSe: index ˜2.5, visible, NIR, MIR, and FIR applications    -   SrTiO3: index ˜2.4, visible and NIR applications, highly        dispersive, persistent photoconductivity    -   CdTe: index ˜2.7, NIR, MIR, and FIR applications    -   Bi4Ti3O12: index ˜2.7, visible applications, electrooptical and        photorefractive    -   BaTiO3: index ˜2.4, birefringent, highly nonlinear,        photorefractive, piezoelectric, pyroelectric, ferroelectric    -   YVO4: index ˜2, birefringent, visible and NIR applications    -   CaWO4: index ˜1.9, slightly birefringent, UV fluorescent, UV,        visible, and NIR applications    -   VO2: phase change material, visible and NIR applications    -   GeSbTe: phase change material, visible and NIR applications    -   In2O3-SnO2 (ITO): index ˜1.7, visible applications, conductive    -   PMMA: index ˜1.5, visible and NIR applications    -   Polycarbonate: index ˜1.6, visible and NIR applications    -   HSQ (spin-on-glass): index 1.4-1.5, visible and NIR applications    -   BaB2O4: ne 1.55, no 1.68, birefringent, UV, visible, and NIR        applications    -   LiB3O5: all refractive index ˜1.6, birefringent, UV, visible,        and NIR bands    -   CsLiB6O10: all refractive index ˜1.45, birefringent, UV,        visible, and NIR applications    -   CdS: solid state laser material and photoresistor    -   PbS: NIR sensitive photoresistor    -   TaAl3: NIR and MIR applications    -   Ta: plasmonic material    -   Be: plasmonic material    -   Pr: plasmonic material    -   Co: plasmonic material    -   Fe: plasmonic material    -   Sn: plasmonic material    -   Nb: plasmonic material    -   Ni: plasmonic material    -   Pb: plasmonic material    -   Pd: plasmonic material    -   Ag: plasmonic material    -   Au: plasmonic material    -   Cu: plasmonic material    -   Al: plasmonic material    -   Na: plasmonic material    -   K: plasmonic material    -   Bi: plasmonic material    -   Pt: plasmonic material    -   Cr: plasmonic material    -   Ti: plasmonic material    -   W: plasmonic material    -   Zn: plasmonic material    -   TiN: plasmonic material.

FIG. 14 illustrates an example device component with four layers 1406-1,1406-2, 1406-3, and 1406-4 of geometric structures. As illustrated, oneor more of the layers includes a unique layout of geometric structures.The device components includes a multi-layer device component.

Experimental/More Detailed Embodiments

In the following discussion, various experimental/more detailedembodiments are described. The skilled artisan would appreciate thatspecific aspects in these detailed examples, although useful inconnection with the previously described embodiments, are not intendedto be necessarily limiting.

In various experimental embodiments, a metagrating is designed using theabove-described optimization process. The metagrating can be designedfor deflecting 75 degrees, in some experimental embodiments. Themetagrating can be fabricated of polycrystalline silicon. For example,polycrystalline silicon (p-Si) is deposited onto multiple wafers,including fused silica substrates and crystalline silicon structureswith a thermal oxide, using silane in a low-pressure chemical vapordeposition furnace at 620° C. The samples consisting of p-Si on thermaloxide on crystalline silicon are optically characterized usingellipsometry to determine the final thickness and the refractive indexof the p-Si. A 30 nm-thick silicon dioxide hard mask is then depositedby plasma-enhanced chemical vapor deposition. The patterns are definedvia electron beam lithography using ZEP 520A resist. The nanostructuresare dry-etched using a breakthrough C2F6 etch, followed by a Cl2, HBr,and O2 main etch, and an HBr and 02 over-etch. The samples are cleanedin a piranha solution, which consists of a mixture of sulfuric acid andhydrogen peroxide heated to 120° C.

FIGS. 15A-15D illustrate examples of characterizing devices designed inaccordance with the present disclosure. To characterize the devices,light from an NKT SuperK white light laser is wavelength-filtered usingan LLTF module, collimated, and polarized. It is then focused onto thedevices using a 0.055 numerical aperture (NA) objective. FIG. 15Aillustrates an example setup for characterizing the device and FIGS.15B-15D illustrate examples of measured angular profiles of the incidentbeam at three different wavelengths. As illustrated by FIG. 15A, thesetup for characterizing the devices can include a polarized halogenwhite light source, coupled to a near-normal dark-filed spectroscopy.For instance, a tunable white light laser can be used as a light source.The angular distribution of the focusing beam is experimentallycharacterized at wavelengths of 1000 nm, 1050 nm, and 1300 nm, and inall three cases, the angular spread is within ±1 degree from normal. Thetransmitted beam is characterized with a germanium power meter mountedon a motorized rotation stage. A section of the substrate with thesilicon fully etched away is used as a calibration window and serves asa reference region for the total transmitted power from the source.

FIGS. 16A-16E illustrate example schematics of a supermode coupling in ametagrating apparatus, in accordance with various embodiments of thepresent disclosure. The dynamics of supermode coupling and propagationin metagratings can be described using coupled Bloch mode analysis(CBMA). CBMA involving the supermodes in grating-like structures can beused to produce in-depth physical insight into the diffraction processin dielectric gratings, and it has more recently been used to explainthe physics of highly reflective deep lamellar gratings. In CBMA, theelectromagnetic fields in the air superstrate above the grating (i.e.,region I in FIG. 16A) and the SiO2 substrate below the grating (i.e.,region III in FIG. 16A) can be expanded by

Fourier harmonics (i.e., Rayleigh or planewave expansion). Theelectromagnetic fields inside the periodic metagrating (i.e., region IIin FIG. 16A) are expanded into a set of N supermodes.

The metagrating can be treated as a vertically-oriented Fabry Perotcavity, in which the supermodes bounce between the air-metagratinginterface and SiO2-metagrating interface (FIG. 16B). As an individualsupermode (indexed as mode j) interacts with an interface, three typesof processes can occur. First, the mode can transmit out of the cavityand couple to the far-field diffraction channels in regions I or III.Second, the mode can reflect from the interface and propagate in theopposite direction in region II. Third, the mode may couple with andexchange energy with the other N-1 supermodes in region II. Thesecoupling dynamics at the top and bottom metagrating interfaces aredepicted in FIGS. 16C and 16D. The interaction of the incident planewave with the bottom metagrating interface shows similar couplingprocesses and is depicted in FIG. 16E.

The supermodes include propagating modes, which possess purelyreal-valued neff (for lossless materials), and evanescent modes, whichpossess very high imaginary-valued neff and exponentially decay alongthe propagation direction. The metagratings presented herein can bethicker than the decay length of the evanescent modes. As such, thepropagating modes are the principal modes responsible for energytransport within the metagrating and for energy funneling into thefar-field diffractive channels. This approximation can be used so longas the evanescent supermodes are effectively damped by the relativelythick metagrating.

Specifically, FIG. 16A illustrates a schematic of a metagratingilluminated by an incident beam through the substrate (dashed-linedsingle-headed arrow), which produces light beams emitted into discretediffraction channels (solid-lined single-headed arrows). FIG. 16Billustrates a dynamic picture of supermode transport and couplingprocesses in the metagrating. FIG. 16C illustrates detailed schematic ofthe mode-coupling processes at the air-metagrating interface for theupward-propagating j^(th) supermode. This mode can couple into thediffracted channels in Region I or couple into downward-propagatingsupermodes (including the j^(th) mode). FIG. 16D illustrates a detailedschematic of the supermode coupling processes at themetagrating-substrate interface for the downward propagating jthsupermode. This mode can couple into the diffracted channels in RegionIII or couple into upward-propagating supermodes (including the j^(th)mode). FIG. 16E illustrates a detailed schematic of the mode-couplingprocesses driven by the incident wave at the metagrating substrateinterface. The incident wave can couple into the diffracted channels inRegion III or into upward-propagating supermodes.

Mathematically, CBMA is derived based on the N supermodes (usuallypropagating modes) bouncing back and forth within the metagrating, whichcan be considered as a Fabry-Perot cavity. The effective mode index ofthe j^(th) supermode can be denoted as n_(j) and its propagation delayover a single trip inside the metagratings φ_(j)=exp(ik_(o)n_(j)h). Theexcitation coefficients of the i^(t)Bloch mode (or n^(th) diffractedorder) can be defined by the j^(th) Bloch mode as r_(ij) ^(top/bot)(ort_(nj) ^(top/bot)) at the top/bottom interface, and the excitationcoefficient of the m^(th) far-field diffracted order the n^(th) order asp_(mn) ^(top/bot). Reciprocity ensures certain equality relations suchas r_(ij) ^(top/bot)=r_(ji) ^(top/bot), t_(ij) ^(top/bot)=t_(ji)^(top/bot) and p_(mn) ^(top/bot)=p_(nm) ^(top/bot). At themetagrating-substrate interface, there can be the relation:

$\begin{matrix}{{\begin{bmatrix}A_{1}^{II} \\A_{2}^{II} \\M \\A_{N - 1}^{II} \\A_{N}^{II}\end{bmatrix} + {R^{bot}\begin{bmatrix}B_{1}^{II} \\B_{2}^{II} \\M \\B_{N - 1}^{II} \\B_{N}^{II}\end{bmatrix}}} = {T^{bot}\begin{bmatrix}A_{0}^{III} \\A_{0}^{III} \\M \\A_{0}^{III} \\A_{0}^{III}\end{bmatrix}}} & {{EQ}\mspace{14mu} 14}\end{matrix}$

With:

$R^{bot} = \begin{bmatrix}{r_{11}^{bot}\varphi_{1}} & {r_{12}^{bot}\varphi_{2}} & \Lambda & {r_{1{({N - 1})}}^{bot}\varphi_{N - 1}} & {r_{1N}^{bot}\varphi_{N}} \\{r_{21}^{bot}\varphi_{1}} & {r_{22}^{bot}\varphi_{2}} & \Lambda & {r_{2{({N - 1})}}^{bot}\varphi_{N - 1}} & {r_{2N}^{bot}\varphi_{N}} \\M & M & O & M & M \\{r_{{({N - 1})}1}^{bot}\varphi_{1}} & {r_{{({N - 1})}2}^{bot}\varphi_{2}} & \Lambda & {r_{{({N - 1})}{({N - 1})}}^{bot}\varphi_{N - 1}} & {r_{{({N - 1})}N}^{bot}\varphi_{N}} \\{r_{N\; 1}^{bot}\varphi_{1}} & {r_{N\; 2}^{bot}\varphi_{2}} & \Lambda & {r_{N{({N - 1})}}^{bot}\varphi_{N - 1}} & {r_{NN}^{bot}\varphi_{N}}\end{bmatrix}$

And

$T^{bot} = \begin{bmatrix}t_{10}^{bot} & 0 & 0 & 0 & 0 \\0 & t_{20}^{bot} & 0 & 0 & 0 \\0 & 0 & O & 0 & 0 \\0 & 0 & 0 & t_{{({N - 1})}0}^{bot} & 0 \\0 & 0 & 0 & 0 & t_{N\; 0}^{bot}\end{bmatrix}$

Similarly, at the top air metagrating interface, there can be therelation:

$\begin{matrix}{R^{top} = {{\begin{bmatrix}A_{1}^{II} \\A_{2}^{II} \\M \\A_{N - 1}^{II} \\A_{N}^{II}\end{bmatrix} + \begin{bmatrix}B_{1}^{II} \\B_{2}^{II} \\M \\B_{N - 1}^{II} \\B_{N}^{II}\end{bmatrix}} = \begin{bmatrix}0 \\0 \\M \\0 \\0\end{bmatrix}}} & {{EQ}.\mspace{14mu} 15}\end{matrix}$

With:

$R^{top} = \begin{bmatrix}{r_{11}^{top}\varphi_{1}} & {r_{12}^{top}\varphi_{2}} & \Lambda & {r_{1{({N - 1})}}^{top}\varphi_{N - 1}} & {r_{1N}^{top}\varphi_{N}} \\{r_{21}^{top}\varphi_{1}} & {r_{22}^{top}\varphi_{2}} & \Lambda & {r_{2{({N - 1})}}^{top}\varphi_{N - 1}} & {r_{2N}^{top}\varphi_{N}} \\M & M & O & M & M \\{r_{{({N - 1})}1}^{top}\varphi_{1}} & {r_{{({N - 1})}2}^{top}\varphi_{2}} & \Lambda & {r_{{({N - 1})}{({N - 1})}}^{top}\varphi_{N - 1}} & {r_{{({N - 1})}N}^{top}\varphi_{N}} \\{r_{N\; 1}^{top}\varphi_{1}} & {r_{N\; 2}^{top}\varphi_{2}} & \Lambda & {r_{N{({N - 1})}}^{top}\varphi_{N - 1}} & {r_{NN}^{top}\varphi_{N}}\end{bmatrix}$

Combining EQ. 14 with EQ. 15 provides:

$\begin{matrix}{\begin{bmatrix}A_{1}^{II} \\A_{2}^{II} \\M \\A_{N - 1}^{II} \\A_{N}^{II}\end{bmatrix} = {\left( {1 - {R^{bot}R^{top}}} \right)^{- 1}{T^{bot}\begin{bmatrix}A_{0}^{III} \\A_{0}^{III} \\M \\A_{0}^{III} \\A_{0}^{III}\end{bmatrix}}}} & {{EQ}.\mspace{14mu} 16}\end{matrix}$

EQ. 16 provides the amplitudes of the upward propagating Bloch modesinside the metagrating, as excited by the external illumination. Theamplitudes of the downward-propagating Bloch modes can be obtained bysubstituting EQ. 16 into EQ. 15. Then the m^(th) diffracted order inRegion I can be written as:

A_(m) ^(I)=Σ_(j)t_(m,j) ^(top)A_(j) ^(II)φ_(j)   EQ. 17.

The amplitudes of the m^(th) diffracted orders in region III can bewritten as:

A_(m) ^(III)=Σ_(j)t_(m,j) ^(bot)B_(j) ^(II)φ_(j)+p_(m0)A₀ ^(III).   EQ.18.

In EQ. 17 and EQ. 18, the far-field diffracted channels are written asthe coherent summation over all the propagating supermodes inside themetagrating, and allow for examination of the supermode interference asthey couple into the diffraction channels. Notably, if only onesupermode is kept in EQ. 17 and EQ. 18, these equations reduce to theclassical Fabry-Perot formulas for a single mode in a periodic medium.

FIGS. 17A-17D illustrate an example metagrating design using anadjoint-based topology optimization. The efficiencies in particularangle metagratings can be understood by examining the optical modes ofthe devices. Metagratings consisting of a single thin film withvertically etched features can be treated as a vertically-oriented FabryPerot cavity supporting supermodes. The substrate-grating andgrating-air interfaces serve as the cavity mirrors. The plane waveincident on the metagrating couples into these modes, which bouncewithin the cavity. Whenever a mode interacts with a cavity mirror, acombination of three processes can occur, as described by coupled Blochmode analysis (e.g., FIG. 18A). First, the mode can reflect from theinterface. Second, the mode can couple with and excite other supermodes.Third, the mode can couple out as plane waves into several discretediffraction channels (six channels are shown in the example in FIG.18A). As such, each diffraction channel contains contributions from allof the supermodes. High deflection efficiency in the desired diffractionchannel is achieved when the out-coupled plane waves from all thesupermodes in that channel (strongly) constructively interfere (FIG.18A, dashed boxes).

FIG. 17A illustrates simulated deflection efficiencies of varioustransmission grating types as a function of deflection angle. Theseinclude the classical echelle grating, three types of establishedmetagrating designs, and topology-optimized metagrating in accordancewith various embodiments disclosed herein. These devices deflectnormally incident TE- and TM-polarized light at a single wavelength, andthe plotted points represent the deflection efficiency averaged overboth polarizations. The wavelengths and materials used in thesesimulations are listed in Table S1. The topology-optimized metagratingcan be optimized for 75 degree deflection.

The grating designs simulated in FIG. 17A can be based on metagratingdesigns in previously published papers. The material systems, operatingwavelengths, and bibliographic sources for these designs are listedbelow:

TABLE 1 Grating Design Approach Material Wavelength Effective MediumTiO₂ 860 Transmit Array Poly-silicon 1550 Geometric Phase TiO₂ 405Echelle (sawtooth profile) SiO₂ 1050

FIG. 17B illustrates a top view of the topology-optimized metagrating,which is optimized for 75 degree deflection in various embodiments. As arepresentative example, the supermodes for the transmissive 75 degreemetagrating featured in FIG. 17B can be analyzed for TM-polarizedincident light. The mode profiles are extracted from a coupled-waveanalysis solver (INSERT 29). This device supports eight propagatingsupermodes, which have effective modal refractive indices ranging fromair-like to silicon-like. The device also supports many evanescentsupermodes, which have decay lengths that are much shorter than thedevice thickness and therefore couple minimally with the diffractionchannels. The field profiles of the propagating modes are plotted inFIG. 18B, and each display an intricate field profile. An analysis ofeach supermode indicates that no individual supermode couplesefficiently into the desired diffraction channel. However, theconstructive interference of all eight out-coupled supermodes in thedesired diffraction channel yields high deflection efficiency. Thedevice deflects normally-incident TE- and TM-polarized light at awavelength of 1050 nm. Black represents silicon and white representsair. FIG. 17C illustrates an example schematic of the forward andadjoint simulations used to optimize a large-angle transmissivemetagrating, which deflects normally-incident light into the +1diffraction channel. Also sketched are the five other diffractionchannels in the system.

FIG. 17D illustrates an example plot of deflection efficiency over thecourse of the adjoint-based topology optimization process for themetagrating in FIG. 17B. A total of 350 iterations can be used to designthe device, although embodiments are not so limited and can include moreor less iterations. The sharp dips represent a geometric blur that isapplied every 40 iterations to eliminate small features (as describedabove). The insets show the dielectric constant distribution in a singleunit cell of the metagrating at different stages of the optimizationprocess.

This example is indicative of the many factors in device design that arecontrolled to enable efficient, and particular angle deflection. First,the effective refractive index and spatial profile of each supermode istailored such that the modes constructively interfere as they coupleinto the intended diffraction channel. Second, the coupling strengths ofthe incident plane wave into the modes; the modes into the diffractionchannels; and the modes with other modes are specified. These couplingparameters involve both the propagating and evanescent supermodes, whichtogether specify the electromagnetic field boundary conditions at themetagrating interfaces. Third, it can be beneficial for the metagratingto support as many total supermodes as possible. Having more modesallows the potential for more degrees of freedom in the design, whichcan be tailored to better fit the parameters above. This design problemis complex and ultimately intractable to address usingphysically-intuitive design procedures, and can be solved using theabove-described optimization process.

In specific experimental embodiments, a transmissive 75 degreemetagrating designed according to the above-described optimizationprocess can be analyzed for efficiencies. An individual grating periodhas space for only two nanoposts, and the device supports only threepropagating supermodes. Two of the modes show strong field localizationwithin an individual nanopost (FIGS. 22A-22C), which is consistent withthe design methodology of stitching together individual waveguideelements. The third has a low effective refractive index, and its fieldsare predominantly in the air region of the grating. Due in part to therelatively small number of supermodes and non-optimized spatial modeprofiles, the grating efficiency is low.

As described above, topology-optimized grating structures operating atnear-infrared wavelengths can be fabricated and characterized. Theinitial substrate is a silicon dioxide wafer, on which a layer ofpolycrystalline silicon is grown by chemical vapor deposition. A 200μm-diameter circular grating devices, in specific experimentalembodiments, is patterned using electron beam lithography, followed byreactive ion etching. To optically characterize the device, a focused,tunable white light laser can be used as a light source, and thediffracted light beams using a germanium detector mounted on agoniometer, as is illustrated by FIG. 15A.

To reduce device sensitivity to fabrication imperfections, therobustness algorithms, as previously described can be implemented intothe design process. These algorithms include the effects of geometricdilation and erosion in each (simulation) iteration of the optimizationprocess, with the goal of reducing the impact of geometric variabilityon device efficiency. This incorporation of robustness into the devicedesign necessitates a tradeoff with optimal device efficiency. Devicesthat possess higher overall performance, at the expense of being lessrobust, can be experimentally realized with more precise fabrication.

The 75 degree transmission grating from FIG. 17B can be characterized,which is designed to deflect normally-incident TE and TM waves with awavelength of 1050 nm. A scanning electron microscopy (SEM) image of thedevice is presented in FIG. 19A, and the silicon nanostructures havemorphologies that match well with the theoretical design. Tilted SEMimages of the device (FIGS. 24A and 24B) show vertical sidewalls,indicative of high-quality silicon etching. From an analysis, both theabsolute and relative efficiencies of the device. Absolute efficiencyrefers to the power in the deflected light beam normalized to the powerof light transmitted through a bare silicon dioxide substrate. Relativeefficiency refers to the power in the deflected light beam normalized tothe total power transmitted through the device.

FIGS. 18A-18C illustrate example supermode coupling, in accordance withvarious embodiments. More-specifically, FIG. 18A illustrates a schematicof mode dynamics in a metagrating device, as described by coupled Blochmode analysis. The supermodes, labeled M_(l) to M_(n), bounce within themetagrating. When these modes scatter at an interface, they can reflect(thick curved arrows), couple with other modes (thin curved arrows), andcouple into diffraction channels (thick straight arrows). Strong beamdeflection occurs when there is strong constructive interference betweenout-coupled modes in the desired diffraction channel (dashed boxes).FIG. 18B illustrates |H| profiles of the supermodes supported by themetagrating designed in FIG. 17B. The incident beam is TM-polarized. Theeffective mode refractive indices neff are defined, and outlines of thesilicon structure are drawn in green. And, FIG. 18C illustrates anexample plot of deflection efficiency as a function of number of modesincluded in the calculation, using the modes in FIG. 18B. When thenumber of modes is one, the diffractive optical properties of only then_(eff)=2.9 supermode is included, and only 16% of the light isdeflected into the desired grating order (inset, top left). As moremodes are included (added in order of decreasing n_(eff)), thedeflection efficiency of the metagrating gradually increases. When alleight modes are included, the deflection efficiency into the desiredchannel is 84% (inset, bottom right).

FIGS. 19A-19F illustrate example experimental characterization ofmetasurfaces, in accordance with various embodiments of the presentdisclosure. FIG. 19A illustrates a scanning electron microscopy image ofthe 75 degree beam deflector shown in 210. FIG. 19D illustrates ascanning electron microscopy of a wavelength splitter fornormally-incident TE-polarized light. Top insets in each of FIGS. 19Aand 19D illustrate magnified images of an individual metagrating unitcell and the bottom insets illustrate a schematic of the metagratingfunction. FIG. 19B and 19C illustrate the theoretical and experimentaldeflection efficiencies of the beam deflector illustrated by FIG. 19A,respectively. Data points are plotted separately for TE- andTM-polarized incidence and are normalized as relative efficiencies. FIG.19E and 19F illustrate the theoretical and experimental deflectionefficiencies of the wavelength splitter illustrated by FIG. 19D,respectively. Data points are plotted separately for 1000 nm and 1300 nmincident wavelengths and are normalized as relative efficiencies. In allplots, values for absolute and relative efficiency are specified.

Theoretical and experimental deflection efficiencies are summarized inFIGS. 19B and FIG. 19C, respectively. The experimental data shows thatthe device operates with high absolute and relative efficiencies. Theabsolute deflection efficiencies for TE- and TM-polarized light aremeasured to be 74% and 75%, respectively, which are close to thetheoretical values. The numerical accuracy of the theoretical values,calculated using an RCWA solver, is benchmarked as previously describedand can have less than a one percent error. The relative efficienciesfor TE- and TM-polarized light are both above 80%, indicating strongpreferential coupling to the +1 diffraction channel, compared to the -1and 0 diffraction channels. The discrepancies between the experimentaland theoretical efficiencies are due in part to minor geometricimperfections in the fabricated device.

The above-described methodology can be applied to multi-functionaldevices. For example, an individual function can be defined to be thedeflection of an incident beam with a particular wavelength andpolarization into a specific diffraction channel. In an experimentalembodiment, a metagrating that deflects 1000 nm TE-polarized light to a+36 degree angle (+1 diffraction channel) and 1300 nm TE-polarized lightto a −50 degree angle (−1 diffraction channel) can be designed,fabricated, and characterized. Each of these target functions areincorporated in a straight-forward fashion into the iterativemetagrating design procedure, by performing forward and adjointsimulations for each function in each iteration. SEM images and theefficiency plots of the device are displayed in FIGS. 19E-F and showthat high-efficiency wavelength splitters can be theoretically designedand experimentally realized. The absolute deflection efficiencies at thetwo target wavelengths are above 60% and are within 10% of theirtheoretical values. The relative efficiencies at these wavelengths arenear 80%.

In summary, the adjoint-based topology optimization as described hereincan be used as an effective method for designing high-performancediffractive optical elements. As a demonstration, in specificexperimental embodiments, these design principles are used to construct75 degree angle silicon metagrating deflectors. These design principlescan readily extend to multi-functional devices. A central feature ofthis approach is the automated specification of a collection ofsupermodes within the metagratings, which possess non-intuitive spatialprofiles and coupling dynamics. Adjoint-based topology optimization inaccordance with the present disclosure can be used to design aperiodic,multi-wavelength, multi-functional metasurfaces with performances thatoperate near the limits of composite nanomaterials engineering.

FIG. 20 illustrates an example mode analysis of a 75 degree beamdeflector designed using the optimization methodology, in accordancewith various embodiments. Specifically, FIG. 20 illustrates theeffective indices neff of the supermodes in the 75 degree deflector forTM polarization. Each circle marker represents a supermode identified bythe RCWA solver. There are 8 propagating modes, contained within thedashed-box box, which have negligible Im(neff). The field profiles ofthese modes are plotted in FIG. 18B. The evanescent modes are containedin the open curve. All of these modes have large imaginary parts anddecay strongly with a propagation length less than or equal to themetagrating thickness.

FIGS. 21A-21C illustrate an example coupled Bloch mode analysis of the75 degree metagrating for TE polarization, in accordance with variousembodiments. FIG. 21A illustrates example effective indices neff of theTE-polarized supermodes of the metagrating, calculated using the RCWAsolver. The device supports five propagating modes, contained within thedashed-lined box, and many evanescent modes, contained within the opencurve. Most of the evanescent modes have large imaginary parts andstrongly decay with a propagation length less than or equal to themetagrating thickness. However, two of the evanescent modes haverelatively small imaginary parts and do not strongly decay within themetagrating. These two evanescent modes, together with the fivepropagating modes, contribute to the deflection efficiency of themetagrating. FIG. 21B illustrates the |E| profiles of the sevensupermodes that contribute to the deflection efficiency of themetagrating. The effective mode refractive indices n_(eff) are listed,and outlines of the silicon structure are drawn in green. FIG. 21Cillustrates an example plot of deflection efficiency of the simulatedmetagrating as a function of number of modes included in the gratingefficiency calculation, for a TE-polarized incident beam. For one mode,the diffractive optical properties of the supermode with n_(eff)=3.0 isincluded, and only 10% of the light is deflected into the desiredgrating order (inset, top left). Note that the total efficiency from alldiffraction channels adds to 100%. As more modes are included (added inorder of decreasing neff), the deflection efficiency of the metagratinggradually increases. When all seven modes are included, the deflectionefficiency into the desired channel is 78% (inset, bottom right).

FIGS. 22A-22C illustrates an example coupled Bloch mode analysis of a75-degree deflector made of two TiO2 square pillars, in accordance withvarious embodiments. In various embodiments, the modes of degreedeflector design can be analyzed based on the effective refractive indexconcept. This design method is represented by the light blue curve inFIG. 17A. For this large deflection angle, the device period is small,such that there is space for only two TiO2 square pillars (see, FIG.22A, inset). This two-pillar design supports 60% and 30% absoluteefficiencies for TM- and TE-incident polarized waves, respectively. Asillustrated by FIG. 22A, coupled Bloch mode analysis for the metagratingfor TM-polarized incident light can be performed and results inverifying that the metagrating supports three propagating modes. Forexample, FIG. 22A illustrates the grating layout and its Bloch modes atTM polarization. FIG. 22B illustrates the |H| of the three propagatingmodes. FIG. 22C illustrates the deflection efficiency as a function ofnumber of modes. For one mode, the diffractive optical properties of thesupermode with n_(eff)=1.61 is included, and 11% of the light isdeflected into the desired direction (inset, top left). As more modesare included (added in order of decreasing n_(eff)), the deflectionefficiency of the metagrating gradually increases. When all three modesare accounted for, the deflection efficiency into the desired channel is60% (inset, bottom right). As shown in FIG. 22B, the field profiles oftwo of the modes (n_(eff)=1.61 and 1.21) are mostly confined in thenanopillars, in agreement with the effective medium approximationapproach. The third propagating mode has an effective index neff ˜1 andits field is mostly distributed in air.

FIGS. 23A-23B illustrates examples of absolute deflection efficiency asa function of different incident angles, in accordance with variousembodiments. The impact of variations in angle of incidence on theperformance of the 75 degree deflector and wavelength splitter from FIG.19A and FIG. 19D. These variations in incidence angle ariseexperimentally because the incident beam is weakly focused on thesample. Plots of simulated absolute efficiency as a function ofincidence angle (θ_(inc)) for both devices are displayed in FIG. 23A andFIG. 23B. For the 75 degree metagrating, as illustrated by FIG. 23A, thecalculation is performed for θ_(inc) ranging from −7 to +1.3 degreesfrom the glass substrate, which corresponds to −10 to +2 degrees in air.The +2 degree angle is a limit that arises from the cutoff of thediffracted beam at larger incident angles. For the wavelength splitter,as illustrated by FIG. 23B, the incident angles range from −7 to +7degrees in glass substrate, which corresponds to -10 to +10 degrees inair. Experimentally, the incident beam has incident angles that rangefrom −1 to +1 degrees in air. As such, the variations in incidence anglemay minimally impact the efficiencies of resulting devices.

FIGS. 24A-24B illustrate examples of tilted scanning electron microscopyimages of metagratings, in accordance with various embodiments. Forexample, FIG. 24A illustrates an image of a 75 degree beam deflector(e.g., FIG. 19A) and FIG. 24B illustrates an image of a wavelengthsplitter (e.g., FIG. 19D). The sidewall profiles of the devices can bevertical.

FIGS. 25A-25B illustrate example plots of experimental metagratingefficiencies as a function of incident wavelength of metagratings, inaccordance with various embodiments. FIG. 25A illustrates an exampleplot of a 75 degree beam deflector (e.g., FIG. 19A) and FIG. 25Billustrates an example plot of a wavelength splitter (e.g., FIG. 19D).

FIGS. 26A-26B illustrate example plots of convergence of metagratings,in accordance with various embodiments. The convergence of the rigorouscoupled-wave analysis (RCWA) simulations can be analyzed. For example,the open-source RCWA code Reticolo can be used as the electromagneticsolver for the optimization process, which Fourier expands theelectromagnetic fields in the system. The numerical inaccuracies of RCWAmainly arise due to inevitable truncation of these Fourier series in thesimulation.

To evaluate the accuracy of the numerical results, the 75 degree blazedmetagrating can be simulated with progressively increasing numbers(N_(x) or N_(y)) of Fourier harmonics in the computation. Here, theFourier series goes from −(N-1)/2 to +(N-1)/2 for each axis. FIGS.26A-26B show the absolute deflection efficiencies of the metagratingwith differing numbers of Fourier harmonics. For example, FIG. 26Aillustrates example results for varying N_(x) values with several fixedN_(y) values and FIG. 26B illustrates example results for varying N_(y)values with several fixed N_(x) values. The vertical cyan lines mark theN_(x) or N_(y) values used in the optimization. The inset in FIG. 26Bshows the metagrating layout under test. The number of Fourier harmonicscan be fixed for one axis and vary the number of Fourier harmonics forthe other axis. FIG. 26A show the convergence tests of absoluteefficiency for varying N_(x) values with several fixed N_(y) values.When N_(y) is large enough (N_(y)>15), all the calculations converge towithin 0.5% of the same value. When N_(y)>15 and N_(x)=27, all of thecalculations produce values that are within 1% of the accurate convergedvalue. Similarly, the simulation results shown in FIG. 26B indicate thatwhen N_(x)>21 and N_(y)=21, all of the calculations produce values thatare within 1% of the accurate converged value. Therefore, simulationswith N_(x)=27 and N_(y)=21 produce results with the numerical accuracieswithin around one percent, and these values can be used for theoptimization process (e.g., the reverse design procedure).

FIG. 27 illustrates an example top view of single period of a 75 degreemetagrating layout, in accordance with various embodiments. The 75degree deflector can be further analyzed using finite-element-method(COMSOL) calculations. COMSOL is an established, commercial softwarepackage that is suitable for modeling curvilinear geometries. Thecurvilinear grating geometry can be approximated using a combination ofsimple geometric shapes and combine them using the “booleans andpartitions” operation in COMSOL. The elementary geometries are delimitedby green and red lines in FIG. 27. The precise geometric dimensions ofthese shapes are recorded in Table 2. The corners of all rectangles arerounded and have a radius of curvature of 8 nm.

The deflection efficiencies of a metagrating of this approximategeometry are calculated using COMSOL to be 84% and 79% for TM and TEpolarizations, respectively, for a normally incident planewave throughthe glass substrate. These values are in agreement with the RCWAcalculations of the metagrating (85% and 79% for TM and TEpolarizations, respectively). This agreement between COMSOL and RCWAsimulations is another indication that RCWA can simulate thecurvilinear, topology-optimized geometries with high accuracy.

As illustrated by FIG. 27, the layout shown in black is used in RCWAsimulations. The approximate layout used in COMSOL consists of acombination of sixteen ellipses and rectangles, which are shown hereoutlined by red and green curves. The dimensions of the grating periodalong x and y are 1087 nm and 525 nm, respectively.

TABLE 2 Geometry Size (nm) Center Position (nm) 1: ellipse Axis length74 × 74 Axis length 74 × 74 2: rectangle Axis length 74 × 74 Axis length74 × 74 3: ellipse Axis length 74 × 74 Axis length 74 × 74 4: ellipseAxis Length 91 × 82.5 (419, 457.5) 5: rectangle Side Length 91 × 74(419, 488) 6: ellipse Axis Length 91 × 82.5 (419, 67.5) 7: rectangleSide Length 91 × 74 (419, 37) 8: ellipse Axis Length 285 × 213 (710,262.5) 9: ellipse Axis Length 285 × 299 (758.5, 262.5) 10: rectangleSide Length 18 × 320 (731, 262.5) 11: rectangle Side Length 19 × 355(749, 262.5) 12: rectangle Side Length 22 × 399 (768, 262.5) 13:rectangle Side Length 20 × 458.5 (788, 262.5) 14: rectangle Side Length52 × 525 (824, 262.5) 15: ellipse Axis Length 85.5 × 458.5 (854.5,262.5) 16: ellipse Axis Length 71 × 299 (884, 262.5)

FIG. 28 illustrates the normalized and absolute efficiencies of a devicecomponent with one layer, two layers, and three layers of geometricstructures across a broadband spectrum. As illustrated, surprisingly,the consistency increases when the number of layers increases.

FIG. 29 illustrates the normalized and absolute efficiencies of a devicecomponent with three layers of geometric structures and another devicecomponent with ten layers of geometric structures across a broadbandspectrum. In various embodiments, with increasing layers, the thicknessof each layer is reduced as the spatial tolerance of silicon is ten nm.As illustrated, the normalized efficiency is relatively the same forthree layers versus ten layers. But, surprisingly, the absoluteefficiency increases for ten layers versus three layers of geometricstructures (given an expectation that the absolute efficiency woulddecrease with the number of layers as silicon, such as poly silicon(e.g., is lossey).

FIGS. 30A-30B illustrate examples of efficiency of a two layer devicecomponent as designed with two thickness. For example, FIG. 30Aillustrates an absolute efficiency and FIG. 30B illustrates a relativeefficiency across a broadband spectrum of 450-750 nm.

FIGS. 31A-31B illustrate examples of efficiency of a three layer devicecomponent as designed with different thicknesses. For example, FIG. 31Aillustrates an absolute efficiency and FIG. 31B illustrates a relativeefficiency across a broadband spectrum of 450-750 nm.

In various specific experimental embodiments, a compact hyperspectralimaging platform that is compatible with existing imaging systemarchitectures is fabricated. To fabricate a periodic or aperiodicapparatus and/or device comprising a plurality of device components(e.g., a metasurface), a series of wafer bonding, alignment, patterning,etching, and planarization steps are used to construct the device layerby layer. For example, in some embodiments, the device components (e.g.,metasurface elements) are integrated directly onto the facet of acommercial CCD platform, together with microlenses and filters, tocomplete optical hardware system. To automate data processing andreconstruction in a fast manner, a custom FPGA electronic backend isused.

FIG. 32A-32B illustrate an example hyperspectral imaging platform thatincludes a periodic or aperiodic apparatus and/or device and opticalproperties of each device components of the device across a broadbandspectrum. The device of the hyperspectral imaging platform is developedto enhance the spectral resolution in hyperspectral imaging systems. Byspecifying quasi-random device components (e.g., metasurface filters)with highly dispersive optical responses over each pixel as illustratedby FIG. 32A, the incident light field is computationally reconstructedand recover edits detailed spectrum (e.g., as illustrated by FIG.33A-33C). These filters are designed to take advantage of spectralsparsity (e.g., there are no sharp spectral peaks in ambient images).

As illustrated by FIG. 32A, each pixel is covered with a differentfilter using the periodic or aperiodic apparatus and/or devices. Lightentering the pixel of the imaging device, such as a conventional CCDcamera, is filtered using the aperiodic apparatus/device with unique andrandom-like spectral characteristics. Light with in a particularwavelength illuminates a particular row of the aperiodic apparatuscorresponding to a row of pixels of the imaging device, as illustratedby the inset of FIG. 32A (e.g., the row of pixels 1423, 1424, 1425,1426, 1427, 1428, 1429, 1430). For example, FIG. 32B illustrates a rowof the periodic or aperiodic apparatus that corresponds to light in the500-530 band (e.g., green). Each pixel (e.g., 1-8) corresponds with aunique layout of geometric structures resulting in optimizedtransmission for a particular wavelength range. FIG. 32B illustrates thedevice components 1433, 1434, 1435, 1436, 1437, 1438, 1439, 1430 andrespective efficiency across a wavelength range that corresponds withthe row of pixels illustrated in the inset of FIG. 32A.

FIG. 33A-33C illustrate an example reconstructed spectra of variousincident waveforms as generated using the high resolution hyperspectralimaging device of FIG. 32A.

FIG. 34 illustrates a plot of different phases of silicon, in accordancewith various embodiments. As illustrated, crystalline silicon hassuperior optical properties at processing blue light as compared topolycrystalline and amorphous silicon. FIG. 34 illustrates therefractive indices for crystalline, polycrystalline, and amorphoussilicon. The solid and dashed lines represent the real and imaginaryparts, respectively, of the refractive index. For each material, thereal part of the refractive indices is high through the visible spectrumand yields the possibility of strong scattering in geometric structuresthat are sub-wavelength in scale.

However, the imaginary parts of the refractive indices are different.Amorphous silicon has a pronounced absorption caused by thedisorder-induced broadening of its bands, which limits its use innear-infrared wavelengths. Polycrystalline silicon has a moderate amountof absorption due to the material disorder at grain boundaries.Crystalline silicon, by contrast, does not have structure disorder whichresults in low absorption below the direct bandgap near 3.4 eV, whichallows the material to be effective as a geometric structure materialacross the full visible spectrum. Further, crystalline silicon iscapable of being etched into geometric structures, such as devicesoperating at blue wavelengths.

In some experimental embodiments, a scattering and absorption of siliconridges are observed. Ridges with rectangular cross-sections are used innano-optical engineering and in individual and coupled arrangements totailor optical responses for metasurfaces. To characterize the differentloss channels in the system, the quasi-normal modes of silicon ridgeswith square cross-sections of varying dimensions are calculated. Thecalculations provide a modal description of the geometric structuresthat comprise dispersive materials. By changing the cross-sectionaldimensions of the ridges, the fundamental mode resonance tunes acrossthe full visible spectrum from 400 nm to 700 nm. The excitation sourceis a plane wave with normal incidence to the ridges, and two incidentpolarizations are analyzed: TM-polarization light, in which the incidentmagnetic field is oriented parallel to the length of the ridge, andTE-polarization light, in which the incident electric field is orientedparallel to the length of the ridge. Calculations are performed foramorphous, polycrystalline, and crystalline silicon using anellipsometry-measured refractive index values illustrated by FIG. 34. Asreferences, calculations are also performed for “ideal silicon,” whichis a non-physical material with the real refractive index of crystallinesilicon but no imaginary part.

FIGS. 35A-35D illustrate a theoretical analysis of the scatteringproperties of silicon ridges. The normalized scattering cross sections(σ_(scat)) of ridges of differing size and composition at theirresonance wavelengths are plotted for TE and TM-polarizations incidencein FIG. 35A and FIG. 35C, respectively. σ_(scat) is the total scatteringcross section divided by the geometric cross section of the ridge.Further, the scattering efficiency is plotted, defined as the scatteringcross section divided by the extinction cross section, for differentridges and excitation conditions in FIG. 35B and FIG.

35D. Starting with TE-polarization, the ideal silicon curve representsan approximate upper boundary for the scattering magnitudes of siliconridges (FIG. 35A) and has scattering efficiency of one-hundred percentsince the material is lossless (FIG. 35B). Its σ_(scat) increasestogether with the refractive index as the mode resonance wavelengthdecreases, as dictated by Mie theory.

The σ_(scat) of crystalline silicon follows that of ideal silicon forwavelengths longer than 500 nm, which is due to its (nearly negligible)imaginary refractive index within this wavelength range (see, FIG. 34).In various embodiments, crystalline silicon represents a nearly ideallossless material at green and red wavelengths. Between 400 nm and 500nm, the absorption in crystalline silicon becomes non-negligible and itsσ_(scat) deviates from that of ideal silicon. Its scattering efficiencygradually reduces within this wavelength range, down to ninety percentat 400 nm. While this represents a reduction in efficiency, it is stillrelatively high. The mode is visualized in the inset of FIG. 35B, whichdisplays |Ez| 2 (out-of-plane component) of a crystalline silicon ridgewith a 27 nm by 2 nm cross-sectional area and eigen-wavelength of450+91i nm. This mode plot indicates that the fundamental TE mode isdelocalized from the silicon ridge, which reduces the mode overlap withthe absorbing silicon ridge and lowers the model loss.

The σ_(scat) and scattering efficiency for amorphous and polycrystallinesilicon indicates that these materials scatter with high magnitude andefficiency at 700 nm, but they perform significantly worse at shorterwavelengths. For amorphous silicon, the scattering efficiency steadilydrops over the visible spectrum down to approximately thirty-fivepercent at 400 nm, which for polycrystalline silicon, the scatteringefficiency drops start at 550 nm to approximately sixty percent at 400nm.

For TM-polarizations, similar trends exist. As with TE-polarization, theσ_(scat) of ideal silicon increases as the fundamental mode resonancewavelength decreases due to the material dispersion of silicon. Theσ_(scat) for crystalline follows that of ideal silicon down to 500 nm,indicating that it represents a nearly ideal lossless material abovethis wavelength. However, its σ_(scat) more precipitously declines from500 nm to 400 nm, and its scattering efficiency decreases fromapproximately one-hundred percent to fifty-percent within thiswavelength range. This efficiency is suitable for many metasurfaceapplications and is high for blue wavelengths. The source of thisreduced scattering, compared to the case of TE-polarization, can beunderstood by examining the fundamental TM mode profile. |Hz| 2(out-of-plane component) is plotted in the inset of FIG. 35D forcrystalline silicon ridge with a 54 nm by 64 nm cross sectional area andan eigen-wavelength of 444+16i nm. Here, the mode is confined in thesilicon ridge, which effectively maximizes mode interaction with thesilicon ridge and enhances modal absorption.

In comparison, the amorphous and polycrystalline silicon ridges havescattering efficiencies that steadily drop throughout the entire visibleregime. For amorphous silicon, the scattering efficiencies are less thanforty percent at wavelengths shorter than 550 nm, and forpolycrystalline silicon, the efficiency drops monotonically toapproximately thirty percent at 400 nm. Both materials possess σ_(scat)that are many times smaller than that of crystalline silicon below 450nm. As such, these materials are limited in their ability to efficientlyscatter blue light due to their material absorption at thesewavelengths.

Although the above discussion is directed to ridges, embodiments inaccordance with the present disclosure are not so limited and caninclude various geometric shapes.

FIGS. 36A-36C illustrate the scattering spectra of individual siliconnanoridges. In various specific experimental embodiments, scatteringfrom crystalline and polycrystalline silicon nano-ridges are analyzed.The crystalline silicon samples are prepared by bonding an SOI wafer toPyrex using hydrogen silsesquioxane (HSQ) as an adhesive layer. Thesilicon handle and buried oxide layer are removed by polishing andetching to expose the crystalline silicon film, which is then patternedby electron beam lithography and etched. Electron microscopy images(FIG. 36A) indicate that crystalline silicon ridges with widths rangingfrom 50 nm to 100 nm can be precisely etched with smooth sideways. Forexample, FIG. 36A illustrates the ectron microscopy images of crystalsilicon ridges with widths of: (i) 50 nm, (ii) 60 nm, (iii) 80 nm, and(iv) 100 nm. The imaged samples are coated with a thin layer of titaniumto minimize charging during imaging, which artificially widens theridges.

The polycrystalline silicon samples are prepared by growing the siliconfilm on silicon dioxide wafer by low pressure chemical vapor deposition,followed by lithographic patterning and etching. Electron microscopyimages of individual ridge are presented and show that etching producedsmooth sidewalls. Films of polycrystalline silicon with thicknesses onthe order of 100 nm or less have relatively small grains, which yieldsmooth sidewall etching.

A polarized halogen white light source, coupled to a near-normaldark-filed spectroscopy setup, characterized light scattering fromindividual 10um-long ridges. Near-normal excitation is used instead ofoblique excitation because it eliminates retardation effects, which candistort the spectral line shapes.

The spectra of 70 nm-thick ridges with differing widths are plotting inFIG. 36B. The spectra show that, as the ridge widths increase from 40 nmto 100 nm, the TM₁₁ fundamental model increases in magnitude and itspeak resonance wavelength shifts from 400 nm to 500 nm. For the 80nm-wide ridge, the TM₁₂ higher-order mode emerges as a distinct featurenear 415 nm and is further pronounced and slightly red-shifted in the100 nm-wide ridge. As such, crystalline silicon geometric structuresexhibit clear and distinct resonant features, including the fundamentaland higher order modes, at blue wavelengths across the 400 nm to 500 nmrange.

Theoretical scattering spectra, calculated using the finite elementprogram COMSOL, are plotted in FIG. 36C and display agreement. Normalincident is assumed, and the simulations account for the glass substrateand finite numerical aperture of the collection objective (NA=0.65). Toconfirm the interpretation of the peak designations, the quasi-normalTM₁₁ and TM₁₂ modes are solved for an 80 nm-widge ridge and there|H_(z)|² intensity profiles are plotted in the set of FIG. 36B. For theTM₁₁ fundamental mode, the eigen-wavelength is solved to be 481 +26i nm,which matches with the peak in the experimental spectra. The modeprofile has the form of a singular lobe confined in the silicon, whichis consistent with the quasi-normal mode analysis. For the TM₁₂ mode,the eigen-wavelength is solved to be 395+11i nm, which matched with theposition of the peak emerging near 400 nm. The mode contains two lobes,confirming its form as a higher order mode.

The experimental spectra of polycrystalline silicon ridges are plottedin FIG. 36B and show distinct peaks across the blue wavelength range.However, the magnitudes of the peaks are lower than that of crystallinesilicon. For 50 nm and 60 nm-wide structures, the peaks have magnitudesthat are four to five times lower than those from crystalline silicon.For 80 nm and 100 nm-wide structures, the peaks have magnitudes that areapproximately half of those from crystalline silicon. In addition, thepolysilicon scattering spectra do not show distinct TM₁₂ pleas in thewider ridges, due to its strong optical absorption. Theses comparisonsin scattering intensity between individual geometric structures confirmthat crystalline silicon is significantly higher quality opticalmaterial at blue wavelengths relative to polysilicon.

The theoretical scattering spectra of polysilicon ridges (FIG. 36C)generally agree with the experimental spectral. However, theexperimental scattering intensity from each ridge is lower compared tothe calculated scattering intensity. This result is attributable tosurface roughness, which is more pronounced in polysilicon than incrystalline silicon due to the presence of grains. Surface roughness canenhance optical losses, contribute to inhomogeneous broadening effects,and scatter light in unpredictable directions. Such roughness is notaccounted for in the theoretical calculations, which assume perfectlysmooth surfaces.

The spectra of ridges excited by TE-polarization do not exhibit clearscattering peaks. A fare-field analysis of back-scattered light fromindividual crystalline silicon ridges indicates that the presence ofhigher order modes at blue wavelengths and their interference with thefundamental mode are responsible for distorting the spectral line shapeof collected light.

Certain more-specific embodiments include a blue light metasurfacedevice. For example, silicon ridges are used as a base element to designand characterize beam deflectors, which are transmissive blazed gratingstructure that deflect light at a specific angle and wavelength. Thesedevice control the magnitude and phase of incident light in ways thatcan generalize to more complex metasurfaces, such as lenses andreflectors. Blue light manipulation plays a role in many technologies.

The following discussion, which focuses on the design of beam deflectorsoperating at three distinct wavelengths, for discussion purposes, it maybe useful to better appreciate the use of such a blue light device. Thefirst is 488 nm, which is a blue light laser line wavelength that iscommon to many fluorescence imaging platforms, including those involvinggreen fluorescent protein. The second is 450 nm, which is the lowerwavelength limit of blue as a color before violet. The third is 405 nm,which is the operating wavelength of Blue-ray optical storage media.

Such apparatuses can also be designed using a stochastic optimizationapproach, which enables the realization of device with designspecification providing such surprising and unexpected opticalresponses. The design deflect TE-polarization light at a twenty degreeangle and utilizing five nanoridges in each period. Based onsimulations, optimal device thicknesses for these wavelengths range from50 nm to 75 nm, which is consistent with the dimensions of ridgespreviously discussed. Further, various apparatuses are designed thatdeflect TE-polarizing light at a forty-five degree angle for eachwavelengths.

FIGS. 37A-37E illustrate example designs and performance of devices. Thetheoretical designs and performance specifications of the beamdeflectors are plotted in FIGS. 37A-37C. The apparatuses operating at488 nm have an absolute efficiency, defined as the total power deflectedinto the desired angle divided by the total incident power through aplain glass substrate, of sixty-five percent. The apparatuses have arelative efficiency, defined as the total power deflected into thedesired angle divided by the total transmitted power, of eight-threepercent. These numbers indicate that metasurfaces with high absolute andrelative efficiencies are realized with crystalline silicon at bluewavelengths. The high relative efficiencies are useful in manytransmissive metasurface applications, where spurious light deflectionleads to distorted wavefronts and noise. The apparatus operating at 450nm and 405 nm do not operate with as high of absolute efficiency as thatoperating at 488 nm due to increased material losses in silicon.However, the absolute and relative efficiencies for 405 nm operation arestill forty-six percent and seventy-six percent, respectively, which aresuitable for many applications.

Various embodiments have been used to experimentally test deflectors.For example, deflectors are fabricated and characterize a 75 nm-thickcrystalline silicon apparatus that deflect TE-polarized 488 nm light bytwenty degrees. The fabrication is consistent with that for individualsilicon ridges. An electron microscopy image of the deflector isillustrated by FIG. 37D and shows a periodic array of spatially definedsilicon ridges. To experimentally test the deflection efficiency, anincoherent white light source is collimated, filtered, and polarizedwith a 488 nm laser line filter, and then the light is focused with a0.14 NA objective lens onto the deflector. The transmitted beam ischaracterized with a power meter mounted on a motorized rotation stageto measure the transmitted power at different angles. The far-fieldprofile of the transmitted light is illustrated by FIG. 37E. The peak attwenty degree deflection is broadened due to the finite numericalaperture of the excitation source. The plot shows that the apparatusdeflects the 488 nm beam at twenty degrees with absolute and relativeefficiencies of sixty-one percent and eighty-two percent, respectively.A comparison between the theoretical (FIG. 37A) and experimental (FIG.37E) deflection efficiencies shows close agreement, indicating thatinhomogeneities intrinsic to the fabrication process minimally impactthe apparatuses.

Embodiments in accordance with the present disclosure can includesilicon ridges and beam deflectors formed of crystalline silicon andthat are suitable for blue light metasurfaces. Polycrystalline siliconmetasurfaces, by contrast, do not scatter blue light as efficiently ascrystalline silicon due to material absorption losses. Blue lightmetasurfaces are useful for a variety of applications in thin-filmoptical engineering. For example, in biology, structured blue lightillumination and focusing is the basis for fluorescence imaging andoptogenetics. Blue light metasurfaces, in various embodiments, enableminiaturized and implantable photonic system for controlled illuminationand imaging. Further, data storage utilizes focused spots of blue lightfor optical storage media. More generally, optical components with bluelight responses are combined with those with green and red responses, orintegrated into broadband designs to yield a broad range of efficientapparatuses responsive to the full visible spectrum.

FIG. 38 illustrates an example of an optimization process, in accordancewith various embodiments. In accordance with a number of embodiments, adevice (e.g., metasurface) is optimized using two processes, the firstis a topology optimization and the second is boundary optimization. Forexample, the topology optimization includes solving Maxwell's equationsand producing an output electromagnetic wave state for a given inputdesign and input electromagnetic wave condition. The boundaryoptimization, in some instances, leverages the simulation engine toproduce a fabricatable (by imposing the fabrication tolerance into theoptimization) design yielding an ideal output electromagnetic wavestate.

A number of embodiments include use of a simulation engine. Examplesimulation engines includes finite-different time-domain (FDTD),finite-difference frequency-domain (FDFD), finite element (FEM), andrigorous coupled wave analysis (RCWA) techniques. The proper choice ofsimulation engine depends on the type of device being designed. Asprevious discussed, the devices are divided into two categories:periodic structures and aperiodic structures. Periodic structuresinclude gratings, and aperiodic structures encompass devices and/orapparatuses that control the magnitude and phase response of light. Forperiodic structures, an open-source RCWA software package provides highcomputational efficiency and numerical stability. For the aperiodicstructures, a combination of aperiodic RCWA (a-RCWA, a modified-versionof RCWA), FDTD (e.g., the commercial software Lumerical), and FEM (e.g.,the commercial software COMSOL), and an open-sourcenear-to-far-field-transformation (NFFT) code can be used. For generaland specific information on simulation engines, reference is made toJ.-P. Hugonin and P. Lalanne. Reticolo software for grating analysis,Institut d'Optique, Orsay, France (2005)(www.1p2n.institutoptique.fr/Membres-Services/Responsables-d-equipe/LALANNE-Philippe);J.-P. Hugonin and P. Lalanne. J. Opt. Soc. Am. A 22, 1844 (2005); and J.Yang, J.-P. Hugonin, and P. Lalanne. ACS Photonics 3, 395 (2016), whichare hereby fully incorporated by reference.

There are many optimization schemes that can be used to optimize anapparatus as previously described herein. In general, one or acombination of optimization schemes are used to yield a hybridoptimization approach to design apparatuses. Examples of optimizationschemes include particle swarm optimization, genetic optimization, localgradient optimization, machine learning approaches using artificialneural network algorithms and deep learning algorithms, and othercomputational approach. For general and specific information on particleswarm optimization, genetic optimization, local gradient optimization,and other approaches, reference is made herein to Jakob S. Jensen andOle Sigmund, Laser Photonics Rev. 5, 308-321 (2011); C. M. Lalau-Keraly,S. Bhargava, O. D. Miller, and E. Yablonovitch. Opt. Express 21, 21693(2013); Jesse Lu and Jelena Vuckovic. Opt. Express Vol. 21, 13351(2013); Y. Zhang, S. Yang, E. J. Lim, G. Lo, T. Baehr-Jones, and M.Hochberg. IEEE Photon. Tech. Lett. 25, 422 (2013); and P. Sanchis, P.Villalba, F. Cuesta, A. Håkansson, A. Griol, J. V. Galan, A. Brimont,and J. Marti. Opt. Lett. 34, 2760 (2009), which are hereby fullyincorporated by reference.

Machine learning approaches using artificial neural network algorithmsand deep learning algorithms are used in various embodiments. While itmay be possible to use a neural network to produce a desired designlayout for a given desired electromagnetic property, these techniquescan also be used in more specialized ways, for example to efficiencyproduce starting points for other optimization schemes. A compellingcompetitive advantage of machine learning approach is that the longerdata is fed to the machine learning algorithm (i.e., days, months, oreven years), the algorithm training gets better and produces betterdevice designs in a more efficient manner.

An example two-step optimization procedure, which may efficiently touchthe globally optimal device design, is now described. As illustrated byFIG. 38, the optimization approach in various embodiments includesinclude an adjoint-based computational approach. Various embodimentsinclude an efficient adjoint-based optimization procedure to design andrefine nano-optical devices that are constrained by practicalfabrication tolerance limits. This route provides an efficientoptimization approach that allows for design of consistently generateddesigns containing thousands of geometric structures and geometricparameters that have performance specifications that approach a globaloptimum without severely increasing computation time. An adjoint-basedapproach consists of two components: topology optimization and boundaryoptimization. Topology optimization determines/sets the number and shapeof elementary geometries (e.g., geometric structures) within a devicelayout, and boundary optimization adjusts (e.g., optimizes) theboundaries of the geometries generated in the topology optimizationprocess. In some embodiments, the topology optimization processeffectively generate geometries that are near the theoretical globaloptimum with high probability. These geometries are then used asstarting points in the boundary optimization routine, which furtherrefines the geometry as well as imposes constraints on minimum featuresize for fabrication purposes. For example, the boundary optimizationcan include adjusting edges between boundaries of the device componentsby accounting for fabrication constraints.

The approach, as illustrated and described herein, provides for a localoptimal solution for a device being optimized that is not convex.However, it is possible to find a local solution that approaches atheoretical global maximum, which will be discussed further herein. Forthis approach, two independent simulations are performed in eachiteration (e.g., in the topology optimization and in the boundaryoptimization): the first is a forward simulation, in which an deviceand/or particular device component is illuminated by the input(s) thatit is designed for, and the second is the adjoint simulation, in whichthe device and/or particular device component is illuminated by itstarget output(s) in the reverse direction. The field distributionscalculated by these simulations allows one to infer the impact of everyparameter on the figure-of-merit (FoM) of the device and/or devicecomponent simultaneously. Specifically, these simulations provideinformation which indicates the effect that a change in any of thegeometric parameters has on the FoM (the FoM gradient for everyparameter). With this information, small modifications are made to thestructure, resulting in a new layout which increases the FoM. Typically,after a small number of optimization iterations, one reaches a localoptimal design. The computational cost of this approach may not scalewith the number of parameters in a device, making it suitable forsystems with large numbers (e.g., many thousands) of geometricstructures.

Due to the number of local optima in such devices, making sure that thesolution is as close as possible to global optimization is difficult. Incertain embodiments, this approach is applied to a number of differentstarting/initial device layouts and used to find the best optimizationresult. Alternatively, various embodiments include identifying abest/optimal (e.g., more suitable) starting point for the optimizationalgorithm by allowing the refractive indices of the materials in thedevice to vary continuously, which yields a more “convex” optimizationproblem (or more technically as having a reduced duality gap). Asolution that satisfies the physical constraints (with the true materialrefractive indices) is obtained by discretizing the continuousrefractive index values, either at the end of the optimization processor gradually throughout the optimization process. In applying thisstrategy towards topology optimization, it is evident that this strategyconsistently yields solutions near the global optimum with highprobability, and is nearly independent of the complexity (in terms ofthe number of layers or geometric structures) of a system.

FIGS. 39A-39M illustrate various examples applications of a device, inaccordance with various embodiments. In various specific aspects, theformed device is used with and/or to form thin film solar cells, ahyper-spectral imaging system, a dielectric lens, a lens, a thermalmanagement metasurface, a light emitting device, a fluorescence imagingsystem, a polarizing lens, a wearable flexible device, and/or amicro-electro-mechanical system (MEM), among other devices and/orsystems.

FIG. 39A illustrates a device (e.g., that is or includes a metasurface)used with and/or to form a thin film solar devices, such as solar cells,photothermal, photocatalytic, solar concentrators, and/or chemicalreaction driving devices. High-efficiency solar cells convert as muchlight as possible into electric currents with semiconductor materials.Semiconductor materials sometimes absorb (and convert) light withefficiencies that strongly vary with wavelength. Optimized geometric(nanophotonic) structures can trap, direct, and/or funnel light inwavelength-specific ways to boost the efficiency of multi-material solarcells. For example, in various embodiments, thin film solar devicesinclude a plurality of device components, each device componentincluding at least one layer of geometric structures. The devicecomponents are configured to capture and sort different wavelengths oflight toward different areas of the thin film solar devices across abroadband range of wavelengths. And, the device components are combinedtogether to form a periodic or aperiodic apparatus and/or device (e.g.,the metasurface).

FIGS. 39B-39C illustrate an example of a device (e.g., that is orincludes a metasurface) used with and/or to form an imaging system. Forexample, as illustrated by FIG. 39B, the device is used with and/or toform hyper-spectral camera, such as a hyper-spectral imaging system (aspreviously illustrated by FIG. 32B). A hyper-spectral imaging systemrecords the spectrum of light at every pixel. This has a very widevariety of applications, from identifying the components anddistribution of a specific material in the environment, enhancingmachine vision, and/or sensing the ripeness of food. Optical elementsare designed that maximally extract spectral information from thesmallest physical footprint on a sensor to realize compact and costeffective hyper-spectral imaging systems. In some embodiments, ahyper-spectral imaging system includes a plurality of device components.Each device component includes at least one layer of geometricstructures and is configured to extract spectral information from asensor of the imaging system across the spectrum of light. Further, thedevice components are combined together to form a periodic or aperiodicapparatus and/or device. In other embodiments, the imaging systemincludes a polarimeter, or both a hyper-spectral imaging system and apolarimeter.

In other specific embodiments, as illustrated by FIG. 39C, the device(e.g., that is or includes a metasurface) is used with and/or to form animaging system, such as an image sensor. For example, the metasurface isintegrated with image sensors to augment their capabilities, reducetheir footprint, and reduce their energy consumption. Image sensorsinclude but are not limited to CCD and silicon CMOS sensors. Thereductions in footprint are due to the device platform, which includesultra-lightweight and nanoscale thin-film form factors. For example, themetasurface can reduce the size of bulky optical imaging systems down tothe footprint of cell phone cameras, which increases the usability andproliferation of advanced imaging systems in the consumer, medical, andmilitary markets. In addition, the ability to integrate multiple opticalfunctions into a monolithic platform produces more durable opticalsystems.

There are a variety of applications of imaging system. For example, animaging system that includes a metasurface in accordance with thepresent disclosure is used for efficient sorting and filtering ofcolors. The imaging system includes, for example, various combinationsof dielectric lens, polarizers, beam splitters, wave plates, phaseplates, filters, mirrors, retroreflectors, beam-shapers, holographicplates, gratings, prisms, or achromatic lenses and/or, deflectors. Aconventional CCD image sensor uses a color filter array that passes red(R), green (G), or blue (B) light to selected pixel sensors. Colorationis achieved by absorbing light in dyes at the undesired wavelengths. Invarious embodiments, flat dielectric devices (or other of the abovelisted components) are used to efficiently focus light of differentwavelength ranges (e.g., different colors) into different pixels andavoid using absorption dyes, which significantly increases thesignal/noise of light entering the sensor. For example, imaging systemin various embodiments includes a dielectric lens that focuses light ofparticular wavelength ranges (e.g., red, green, and/or blue light, thevisible spectrum, near-infrared, and/or ultraviolet light) to selectedpixel sensors. In some embodiments, the dielectric lens includes aplurality of device components formed by at least one layer of geometricstructures, and configured to focus red, green, and/or blue light toparticular pixel sensors. Further, the device components are combinedtogether to form a periodic or aperiodic apparatus and/or device.

Various embodiments include using the apparatus (e.g., that is orincludes a metasurface) for energy efficient optics. For example, themetasurfaces are optimized to complement computational imaging softwareto optimize for energy consumption in an optical system. As such, anadvanced optical system is created that not only have cell phone cameraform factors, but that can also run off of a cell phone battery.Applications include compact microscopy, hyperspectral, polarimetry,light field camera, holographic display, and sensing systems.

FIGS. 39D-39E illustrate various examples of using an apparatus (e.g.,that is or includes a metasurface) to form lenses. In some applications,such as machine vision systems, systems use near-infrared light sourcesto illuminate the environment. Flat optics devices are used to controlthe structural illumination patterns and their phase, focus and processincoming/scattered light, and perform optical processing and filteringin optical hardware at one or more wavelengths with high efficiency.

In some specific embodiments, the device (e.g., that is or includes ametasurface) is used with and/or to form a flat lens. For example, dueto the material dispersion, classic lenses exhibit chromatic aberration,which means that light of different color are focused to different focalspots (chromatic aberration). In addition, classic lenses are incapableof perfectly focusing light which is incident from all angles (coma).These facts prevent high-quality imaging and are sometimes solved bycombining multiple lenses together to form a bulky optical system. Invarious embodiments, ultra-thin (several hundred nanometers thick)metasurface flat lenses are designed to eliminate both chromaticaberration and distortion. This allows for the creation of ultra-compacthigh-quality optical systems, and reduces the physical size andincreases the image quality of cameras in compact consumer devices. Flatlenses are also designed, at single or multiple wavelengths, with verylarge numerical aperture (0.9+ in air, 1.4+ in oil). Metasurfaces, inaccordance with the present disclosure, are used to design and makelarge aperture gradient index lenses. The extension of the metasurfaceto infrared wavelengths also enables compact and advanced imagingmodalities at these wavelengths. For example, in a number ofembodiments, a lens includes a plurality of device components, eachdevice component including at least one layer of geometric structuresand is configured to focus particular wavelengths of light across abroadband spectrum to different focal spots and incident from allangles. The device components are combined together to form a periodicor aperiodic apparatus and/or device.

In various other specific embodiments, the device (e.g., that is orincludes a metasurface) is used with and/or to form a thermal radiationmanagement metasurface. For example, the device is tailored to havethermal radiation properties (e.g., emissivity, spectrum anddirectionality of emission), which has applications as thermal sources,in thermal imaging, countermeasures, and in applications pertaining tothermophotovoltaics. In some embodiments, a thermal radiation managementmetasurface changes the temperature of a volume surrounded by themetasurface. The metasurface includes a plurality of device components,and each device component includes at least one layer of geometricstructures and having particular radiation properties (e.g., emissivity,spectrum and directionality of emission). Further, the device componentsare combined together to form the metasurface.

FIG. 39F illustrates an example of using a device (e.g., that is orincludes a metasurface) with and/or to form a light emitting device,such as high density light field displays, laser, diodes light bulb, LEDand/or OLED (and/or other general light emitting device). The lightextraction efficiency from light emitting devices is limited by thetotal internal reflection and subsequent absorption of light. Theintegration of a metasurface as described herein directly with lightemitting device enhances extraction efficiency. This reduces theoperating temperature of such devices and greatly increase theefficiency of light emission. More generally, the device components ofthe metasurface serve as optical impedance matching elements withangular and wavelength dependence to enhance or reduce the transmissionof light in novel ways. In various embodiments, a light emitting deviceincludes a thin metasurface attached to a window of the light emittingdevice that is configured and arranged to allow light to escape to freespace. The thin metasurface includes a plurality of device components,each device component including at least one layer of geometricstructures and optimized to steer the light to specific directions withspecific polarizations, to filter the light, to increase extractionefficiency and minimize energy absorbed by the light emitting device toreduce the operating temperate and increase efficiency of lightemission. Further, the device components are combined together to formthe thin metasurface.

FIGS. 39G-39H illustrate examples of using a device (e.g., that is orincludes a metasurface) with and/or to form a component of a microscopyapparatus. In a number of embodiments, the metasurface is tailored toserve as components in high-performance and miniaturized microscopysystems. In particular, the metasurface and/or device components combinelensing, beam steering, beam splitting, filtering, and magnificationfunctionality in ways that are tailored to the wavelength andpolarization of light. In some embodiments, the metasurfaces are used toencode specific phase patterns as a function of wavelength, which can beused in diverse applications ranging from structural light imaging tosingle molecule imaging. In other embodiments, the metasurfaces achievebeam expansion and contraction within a very small path length. Forfluorescence imaging, the metasurface manipulates incident andfluorescent light in different ways to minimize system form factor andboost signal to noise ratio.

Such devices are used in a broad range of microscopy platforms includingbut not limited to: fluorescence imaging, phase contrast imaging,single-molecule imaging, light-sheet microscopy, and others. Forexample, some embodiments include a fluorescence imaging, phase contrastimaging, single-molecule imaging, light sheet microscopy, and/or othersuch systems that contain a polarizing lens, the lens including aplurality of device components. Each device component includes at leastone layer of geometric structures and configured to sense polarizationacross a wavelength range (e.g., broadband or different range) of light,wherein the device components are combined together to form a periodicor aperiodic device and/or apparatus.

In some embodiments, a device (e.g., that is or includes a metasurface)has different optical responses and/or properties for different rangesof light. Such example devices have optical responses that change as afunction of wavelength or polarization of light (e.g., performwavelength and polarization multiplexing). As a specific example, adevice differently modulates light of a first wavelength range (e.g.,green light) than a modulation of light of a second wavelength range(e.g., red light). In another specific example, a lens focuses lightdifferently at different wavelengths. In various embodiments, suchdevices (e.g., spatial filters) can change the shape of light as afunction of wavelength and polarization of light and, thus, havemultiple functions (for different wavelengths) as the devicesrespond/modulate light differently for different wavelengths.Accordingly, some embodiments include devices that have differentoptical functionality at different wavelengths. Such devices are usefulin areas of compressive imagine sensing, remote sensing, opticalmicroscopy, etc. to enhance resolution, increase speed of acquisition,etc.

FIG. 39I illustrates an example of using a device (e.g., that is orincludes a metasurface) with and/or to form an imaging device. Forexample, the metasurfaces are used to produce high quality thin filmlenses that are capable of withstanding temperatures well above thoseused in the reflow process. This makes metasurface lenses an idealcandidate for advancing both the miniaturization of high performancecameras as well as the automation of the optical system assemblyprocess.

A major hindrance to the full automation of camera modules for consumerapplications is that the materials used for the lenses themselves arenot capable of surviving high heat environments. This means that thosecomponents are not compatible with the reflow process, the process bywhich electronic components are joined together simultaneously withsolder. Various embodiments include an imaging device that includes atleast one lens. The at least one lens including a plurality of devicecomponents, each device component including at least one layer ofgeometric structures and having particular optical properties for aparticular optical response, wherein the device components are combinedtogether to form a periodic or aperiodic device and/or apparatus.Further, the periodic or aperiodic device/ apparatus (e.g., metasurface)is joined together with other electronic components of a camera modulevia soldering.

FIG. 39J illustrates an example of using a device (e.g., that is orincludes a metasurface) with and/or to form a wearable flexible device,such as a lightweight wearable optical system. For example, themetasurface optimization scheme is used to design thin film opticaldevices that are mounted onto flexible substrates. The metasurfaces areintrinsically mechanically flexible due to their thin film form factor.As such, they can be directly integrated into wearable devices such asclothing, contact lenses, prescription glasses, goggles, watches, andflexible medical devices. Some specific embodiments include a wearableflexible device that includes a plurality of device components. Eachdevice component includes at least one layer of geometric structures andhas particular optical properties for a particular optical response,wherein the device components are combined together to form a periodicor aperiodic device and/or apparatus. In some embodiments, theflexibility of the device scales in relation to the thickness of theperiodic or aperiodic device and/or apparatus. In other specificembodiments, the metasurface is used to form solar cells for a wearableflexible device. For instance, as described above, the metasurface isused to form the solar cells and integrated into a wearable flexibledevice to provide a renewable energy source (e.g., solar energy) for thewearable flexible device.

FIG. 39K illustrates an example of using a device (e.g., that is orincludes a metasurface) with and/or form a nano-electro-mechanicalsystem (NEMS) and/or a micro-electro-mechanical system (MEM). Forexample, silicon-based metasurfaces are actuated using concepts in NEMSand/or MEMS to enable a new class of high performance optical MEMS.These include NEMS and/or MEMS using conventional hard materials usingwafer-based fabrication, and also mechanically compliant soft NEMSand/or MEMS utilizing mechanically soft materials such as elastomers andpolymers. Applications include miniaturized and high speed beam steeringplatforms, scanners, lenses with adjustable zoom, optical force sensingplatforms, and spectrometers. These devices can be implemented on solidstate substrates or mounted on active and passive devices with unusualform factors such as the ends of an optical fiber or laser facet. Anumber of embodiments includes a NEMS or MEMS including at least onemetasurface. The metasurface includes a plurality of device components,each device component including at least one layer of geometricstructures and having particular optical properties for a particularoptical response, wherein the device components are combined together toform the metasurface

Other specific embodiments, include using a device (e.g., that is orincludes a metasurface) in lightweight optics for deployment into spaceor high speed aircraft. The metasurface is useful for optics in avionicsapplications because they are lightweight, conformal to flat and curvedgeometries, and are monolithic and therefore mechanically. Possible usesin such systems include but are not limited to retroreflectors, cameraobjectives, photodetector objectives, beam steering, telescopeassemblies, or solar cell coatings mounted on high speed aircraft suchas commercial jets, military jets, drones, and spacecraft.

FIG. 39L illustrates an example of using at least one device (e.g., thatis or includes a metasurface) in a three-dimensional (3D) display. Bycontrolling the amplitude and phase of red, green, and blue light,metasurfaces produce a fully visible holographic image. Devices can bealso be designed to integrate with 3D imaging displays to producecompact 3D displays. These concepts can apply to heads-up displays,virtual reality, augmented reality, and glassless 3DTV.

Further, specific embodiments include use of device (e.g., metasurface)to form an ultra-high resolution imaging system. Using the optimizationapproach, geometric structures are designed in various embodiments withoptical responses that span the full 1931 CIE color spectrum. Thesegeometric structures can serve as building blocks for ultra-highresolution (100,000+ pixels/inch) imaging systems with full colorresponse. These devices can also be combined with active light emissiondevices, such as LEDs, to produce vivid and precise colors beyond thescope of current red/green/blue coloration technologies.

A number of embodiments include use of device (e.g., that is or includesa metasurface) to form and in compact quantum information platforms. Thedevice, which can coherently manipulate light in multiple ways (i.e.phase, amplitude, and polarization states) with ultra-high efficiency(99%+ in the near-infrared), can enable ultra-compact optical setups forquantum computing. For example, a metasurface manipulates individualphotons with extremely high fidelity, process entangled photon pairswith minimal loss, and steer and guide photons with high efficiency, andproduce squeezed states. The integration of active photonic materialsinto the devices, such as quantum dots and quantum wells, can furtherstreamline the solid state integration of quantum optics elements in asingle monolithic platform.

FIG. 39M illustrates an example of using a device with and/or to formcomponent of a polarimeter. Some embodiments include the use of thedevice to form a compact broadband metasurface polarimeter. Traditionalpolarimeters use bulky optical components to sort the light bypolarization state. Metasurfaces, in some embodiments, are designed toproduce a broadband response that sorts light of different polarizationstates. Further, metasurfaces are designed to convert incident light ofa polarization and wavelength type into a distinct spatial pattern,which can be detected by a conventional CCD. Such concepts are used toconstruct compact hyperspectral polarimeter imaging sensors, to recordan amount of information that far exceeds that of conventional imagesensors, and/or for applications in machine vision, scientific metrologyexperimentation, environmental sensing, and military application.

Various embodiments include using device (e.g., that is or includes ametasurface) to form nonlinear optical elements. For example, non-linearoptical materials are integrated into the metasurface to produce systemsthat: 1) have optical responses that depend on incident light intensity,2) have optical responses that depend on local electric field, and 3)produce output light responses with wavelengths differing from the inputlight. Such systems have applications in actively electrically oroptically controlled photonic devices for steering, sorting, andfiltering light in free space and on chip; compact quasi-phase matchingdevices (e.g., efficient conversion of wavelengths in a compact formfactor), one way mirrors, and/or compact mode locking systems.

Other specific embodiments include use of device (e.g., that is orincludes a metasurface) in and/or to form transformation optics andunusual beam manipulation platforms. The ability for metasurfaces todeflect light in arbitrary ways is used to optimize for light deflectionand steering in the context of transformation optics and negativerefractive index materials. Such a capability is useful in applicationssuch as invisibility cloaking, routing and waveguide devices,ultra-compact active and passive optical cavities, sensors, imagingarrays, and solar light funnels.

In accordance with various embodiments, a device (e.g., that is orincludes a metasurface) is used in and/or to form active and passiveoptical cavities. For example, the metasurface serves as facets inoptical cavities, and in general, in applications requiring multiplepasses of light. This is particularly the case in the near- andmid-infrared wavelengths, where metasurfaces possess negligibleabsorption losses. Metasurfaces in some embodiments are implemented inlaser cavities to direct/specify feedback, mode profiles, mode spacing,mode volume, system form factor, and mediate nonlinearities in pulsed orCW systems. This applies for laser types including but not limited tofree space lasers, semiconductor lasers, ultra-fast lasers, pulsedlasers, SBS lasers, and plasmonic lasers. It can also be implemented inpassive optical cavities to modify the spatial form factor, effectivecavity size, sensitivity, polarization modes, and angular momentum modesof the system.

Further, in various specific embodiments, a device (e.g., that is orincludes a metasurface) is used in and/or to form dynamic devices.Notably, materials with optical properties that are reconfigured can beincorporated directly into the metasurfaces to reconfigure their opticalresponse. These include, as examples, phase change materials (such asGST and vanadium dioxide), which have optical responses that can bemodified as a function of electrical, mechanical, or thermal stimulus,electro-optic materials (such as bulk semiconductors/thin filmsemiconductors/2D materials/transparent conductive oxides with tunablecarrier concentrations; electrochemical materials that reconfigure as afunction of chemical composition; liquid crystal tuning, magnetic fieldtuning, and MEMS tuning). As such, devices can be made that candynamically steer or route light on chip and in free space, change focalpoint position in the case of a lens, serve as dynamically tunablefilters/modulators/phase plates, dynamic holographic surfaces, etc.Further, devices can be integrated with active on-chip devices, such asmodulators, detectors, and sources, to produce systems with hybridoptoelectronic capabilities.

It may also be helpful to appreciate the context/meaning of thefollowing terms. Geometric structure refers to or includes a materialhaving a geometric shape and/or size defined by same-wavelength and/orsub-wavelength dimension(s), such as in a resonator and/or as in adevice or component with geometry-dependent optical properties. Ageometric structure can be a photonic structure or element used forproviding refractive or reflective properties. In some embodiments, thegeometric structure can be formed of at least one (low absorption)dielectric or semiconducting material, such as materials having arefractive index that is greater than two and/or materials that are notprimarily metallic. As an example, the geometric structures can includea pattern etched into a dielectric or semiconducting material. Thegeometric structures are designed using the above-describediterative-based topology optimization and are not constrained to aspecific topology. Nanostructure refers to or includes a geometricstructure with at least one dimension across the structure that is lessthan a micron. Device component refers to or includes at least one layerof geometric structures, where one or more layers include a geometriclayout of one or more geometric structures and that are stacked to formthe device component. Each device component can be a dielectric orsemiconducting film(s) arranged on the flat substrate (e.g., each of theplurality device components are arranged on the flat substrate). Theplurality of device components forming the apparatus can support aplurality of optical modes which include inter-mode and intra-modecoupling that is mediated by the bouncing of light between differentvertical interfaces of the layers of the device components. In variousspecific embodiments, the plurality of device components provide atleast 3 round trips of light bouncing. Example device components caninclude lens, polarizers, beam splitters, wave plates, phase plates,filters, mirrors, retroreflectors, beam-shapers, holographic plates,gratings, prisms, or achromatic lenses and/or, deflectors, among otheroptical elements. Layer refers to or includes a material with a regionthereof formed uniformly or with a thickness of one or more materials asexemplified in a device component having multiple layers forming athickness dimension of the device component. Microstructure refers to orincludes a structure, such as in a device component, with at least onedimension that is of a magnitude order of a micron. Periodic deviceand/or apparatus refers to or includes a device and/or apparatus (havingmultiple device components as above) formed of periodic structures, suchas device components with geometric structures of a periodic pattern(e.g., device components with regularly arranged layers). For example, aperiodic device and/or apparatus is configured to modulate light in aperiodic pattern. Aperiodic device and/or apparatus refers to orincludes a device and/or apparatus formed of aperiodic structures, suchas devices components formed of different geometric structures in anaperiodic pattern. Metasurface refers to or includes optical hardwareand/or a device that is configured to control a magnitude and phaseresponse to light based on geometric design of the metasurface (e.g.,film, rigid or flexible, and/or other insulative/(semi-)conductivesubstance providing desired optical properties). Optical propertiesrefers to or includes an interaction with electro-magnetic radiation oflight. Examples of optical properties include reflection, refraction,diffusion, absorption, and transmission, which can include specificangles. Optical response refers to or includes optical properties and/orrespective efficiencies at particular wavelengths of light. Topology ofa device component refers to or includes a layout of geometricstructures in each layer of the device component (e.g., topologyincludes various parameters such as the number of layers, the layout ofgeometric structures in each layer, the shape of the geometricstructures, dimensions of geometric structures, dimensions of eachlayer, include thickness, among other parameters). Continuous profilerefers to or includes a topology of the device component having a rangeof two or more materials and including mixtures of the two or morematerials. Broadband spectrum refers to or includes a range ofwavelengths, an example broadband spectrum includes the visible lightspectrum, near-infrared spectrum, infrared spectrum, and a combinationthereof. Spectral information refers to or includes information relatingto or produced by a spectrum of light. Discrete profile refers to orincludes a topology of the device component having two or more materialthat is discrete and does not include mixtures of the respectivematerials, such as the discrete profile being generated by convertingthe continuous profile to a binary result. And fabrication constraintsrefers to or includes parameters indicative of fabrication tolerancelimits (example parameters include constraints on minimum geometricstructure dimensions for fabrication purposes).

As should be readily apparent, various modules and/or othercircuit-based building blocks may be implemented to carry out one ormore of the operations and activities described herein and/or shown inthe block-diagram-type figures. In such contexts, these modules and/orbuilding blocks represent circuits that carry out one or more of theseor related operations/activities. For example, in certain of theembodiments discussed above, one or more modules and/blocks are discretelogic circuits or programmable logic circuits configured and arrangedfor implementing these operations/activities, as in the circuitmodules/blocks therein. In certain embodiments, the programmable circuitis one or more computer circuits programmed to execute a set (or sets)of instructions (and/or configuration data). The instructions (and/orconfiguration data) can be in the form of firmware or software stored inand accessible from a memory (circuit). As an example, first and secondmodules/blocks include a combination of a CPU hardware-based circuit anda set of instructions in the form of firmware, where the first module/block includes a first CPU hardware circuit with one set of instructionsand the second module/block includes a second CPU hardware circuit withanother set of instructions. Also, although aspects and features may insome cases be described in individual figures, it will be appreciatedthat features from one figure or embodiment can be combined withfeatures of another figure or embodiment even though the combination isnot explicitly shown or explicitly described as a combination.

Consistent with aspects of the foregoing description, a number ofrelated aspects and embodiments are disclosed in accordance with theAttachments, entitled Attachment A, Attachment B, and Attachment C,which are fully incorporated herein by reference. For example,Attachment A illustrates various embodiments and aspects as previouslydescribed herein, such as aspects previously described by FIGS. 2A-2C,15A-15D, 16A-16E, 17A-17C, 18A-18C, 19A-19F, 20, 21A-21C, 22A-22C, 24A,25A, 26A-B, and 27, among other locations. Attachment B illustratesdevices designed using the above-described topology optimization. Forexample, as illustrated by FIGS. 3a-3d of Attachment B, thetopology-designed device can support many optical modes (e.g., sevenmodes), and which include inter-mode and intra-mode couplings due tobouncing of light within the device components. By contrasts, FIGS.2a-2d illustrates a conventional optical device (e.g., bulk optics orother metasurfaces designed using other techniques) in which lightentering the device comes out in a single path (e.g., does not bouncewithin the device/does not include inter-mode couplings). The whitesquares illustrated in FIGS. 2d and 3d represent optical modes that donot bounce or couple with each other (e.g., light goes in and comesout). As illustrated, the device designed using thetopology-optimization method described herein (e.g., FIG. 3d ) support agreat number of optical modes than the conventional device (e.g., FIG.2d ). FIG. 4a-e of Appendix B illustrates a device that supports around50 modes. In specific embodiments, apparatuses formed can support atleast three round trips of bouncing within device components and whichrecovers the steady state performance of the device, as reflected inFIGS. 3d and 4d of Appendix B. Attachment C illustrates differentexample devices as designed to support different function using thetopology-optimization method. For example, FIG. 2 of Attachment Cillustrates a device that transmits and deflects light through thedevice into different directions depending on the wavelength of thelight. FIG. 3 of Attachment C illustrates experimental efficiencies ofdifferent devices, which can be defined as N^(−1/4) in variousembodiments and as further illustrated by FIG. S3. For instance,embodiments herein and/or in the provisional application (including theslides therein) may be combined in varying degrees with the embodimentsillustrated in the Attachments (including wholly).

Various embodiments are implemented in accordance with the underlyingProvisional Application (Ser. No. 62/329,841), entitled “DeviceComponents Formed of Multiple Layers of Geometric Structures or a ThreeDimensional Multi-layer, Multi-Material Metasurface”, filed Apr. 29,2016, to which benefit is claimed and which are fully incorporatedherein by reference. For instance, embodiments herein and/or in theprovisional application (including the slides therein) may be combinedin varying degrees (including wholly). Reference may also be made to theexperimental teachings and underlying references provided in theunderlying provisional application, including the slides that form partof the provisional application. Embodiments discussed in the slides arenot intended, in any way, to be limiting to the overall technicaldisclosure, or to any part of the claimed invention unless specificallynoted.

Various embodiments include the use of such device components in anoptical-type device. For example, various specific embodiments aredirected to an apparatus that is a thin film solar device which includesthe plurality of device components, each device component including atleast one layer of geometric structures and configured to capture andsort different wavelengths of light toward different areas of the thinfilm solar device across a broadband range of wavelengths. In otherembodiments, the apparatus is a hyper-spectral imaging system configuredto record a spectrum of light at each of a plurality of pixels, thehyper-spectral imaging system including the plurality of devicecomponents, each device component configured to extract spectralinformation from a sensor of the imaging system across the spectrum oflight. Further, the apparatus can be an imaging system including adielectric lens configured to focus light of particular wavelengthranges to selected pixel sensors, wherein the dielectric lens includesthe plurality of device components configured to focus light ofparticular wavelength ranges to particular pixel sensors. In addition oralternatively, the apparatus is a lens including the plurality of devicecomponents, each device component configured to focus particularwavelengths of light across a broadband spectrum to different focalspots and incident from all angles. In some specific embodiments, theapparatus is a thermal radiation management metasurface configured andarranged to change a temperature of a volume and/or area surrounded bythe metasurface, wherein the metasurface includes the plurality ofdevice components, each device component having particular radiationproperties. In other embodiments, the apparatus is a light emittingdevice including a metasurface attached to a window of the lightemitting device that is configured and arranged to allow light to escapeto free space, the metasurface including the plurality of devicecomponents, each device component optimized to steer the light tospecific directions with specific polarizations, to filter the light, toincrease extraction efficiency and minimize energy absorbed by the lightemitting device to reduce an operating temperate and increase efficiencyof light emission.

In various embodiment, the apparatus is an imaging system have apolarizing lens, the lens including the plurality of device components,each device component configured to sense polarization across awavelength range of light, wherein the device components are configuredand arranged to provide lensing, beam steering, beam splitting,filtering, and/or magnification functionality. For example, theapparatus can be an imaging device including at least one lens bonded toa portion of the imaging device, the at least one lens including theplurality of device components, each device component having particularoptical properties for a particular optical response.

In other specific embodiment, the apparatus is a wearable flexibledevice including the plurality of device components, each devicecomponent having particular optical properties for a particular opticalresponse, wherein a flexibility of the wearable flexible device scalesin relation to a thickness of the wearable flexible device.Alternatively and/or in addition, apparatus is a NEMS or MEMS, includingat least one metasurface formed by the plurality of device components,each device component having particular optical properties for aparticular optical response.

Terms to exemplify orientation, such as upper/lower, left/right,top/bottom, above/below, width, length, and thickness (as well as x, y,and z), may be used herein to refer to relative positions of elements asshown in the figures. It should be understood that the terminology isused for notational convenience only and that in actual use thedisclosed structures may be oriented different from the orientationshown in the figures. Thus, the terms should not be construed in alimiting manner.

Certain embodiments are directed to a computer program product (e.g.,nonvolatile memory device), which includes a machine orcomputer-readable medium having stored thereon instructions which may beexecuted by a computer (or other electronic device) to perform theseoperations/activities.

Various embodiments described above may be implemented together and/orin other manners. One or more of the items depicted in the presentdisclosure can also be implemented separately or in a more integratedmanner, or removed and/or rendered as inoperable in certain cases, as isuseful in accordance with particular applications. In view of thedescription herein, those skilled in the art will recognize that manychanges may be made thereto without departing from the spirit and scopeof the present disclosure.

Based upon the above discussion and illustrations, those skilled in theart will readily recognize that various modifications and changes may bemade to the various embodiments without strictly following the exemplaryembodiments and applications illustrated and described herein. Suchmodifications do not depart from the true spirit and scope of variousaspects of the disclosure, including aspects set forth in the claims.

What is Claimed:
 1. A method comprising: geometrically optimizing aperiodic or aperiodic device comprising a plurality of devicecomponents, each device component including at least one layer ofgeometric structures, by optimizing a topology, for each devicecomponent, from a starting point to have particular optical propertiesfor a particular optical response including: selecting the startingpoint for a continuous profile to have the particular optical propertiesfor the particular optical response; iteratively converging thecontinuous profile to a discrete profile; and while iterativelyconverging to the discrete profile, adjusting edges between boundariesof the device components by accounting for fabrication constraints. 2.The method of claim 1, further including determining an optimizedstarting point for a topology of one or more of the device componentsincluding at least one layers of geometrical structures, configured tohave particular optical properties for a particular optical response. 3.The method of claim 1, wherein each geometric structure includes or ageometric shape and size defined by same-wavelength and/orsub-wavelength dimensions, and having optical responses that change as afunction of the wavelength or polarization of light.
 4. The method ofclaim 1, wherein selecting the starting point for the continuous profileincludes providing a topology for the device components that includesthe continuous providing having a random dielectric continuum ofdielectric constants ranging between air and silicon.
 5. The method ofclaim 1, wherein iteratively converging the continuous profile to thediscrete profile includes iteratively simulating two electromagneticsimulations to produce two sets of electromagnetic field profiles andadjusting the continuous profile toward the discrete profile using theproduced two sets of electromagnetic field profiles during eachiteration.
 6. The method of claim 1, wherein converging the continuousprofile to the discrete profile during each iterations includesimproving a Figure of Merit (FoM) by changing a dielectric constant atone or more locations of the device components and over a plurality ofiterations to cause a dielectric continuum of the device component atlocations to converge to the dielectric constant of materials formingthe geometric structures.
 7. The method of claim 1, wherein iterativelyconverging the continuous profile to the discrete profile furtherincludes performing a plurality of iterations of a forward simulationand adjoint simulation to specific changes in a dielectric constant ateach location of one or more device components that improves a Figure ofMerit (FoM).
 8. The method of claim 7, further including, over theplurality of iterations, converging a dielectric continuum in the devicecomponents to a dielectric constant of silicon or air.
 9. The method ofclaim 1, wherein adjusting the edges between boundaries of the devicecomponents further includes periodically adjusting the edges betweenboundaries of the device components by accounting for fabricationconstraints during the iterative converging the continuous profile tothe discrete profile.
 10. The method of claim 1, wherein adjusting theedges between boundaries of the device components further includessetting a space between the device components to mitigate or minimizecoupling between adjacent device components and/or to cause anapproximate linear phase profile response.
 11. The method of claim 1,further including geometrically optimizing the periodic or aperiodicdevice for having particular optical properties for a plurality ofoptical responses including converging the continuous profile to thediscrete profile by performing forward and adjoint simulations for eachof the plurality of optical response in each of a plurality ofsimulation iterations.
 12. An apparatus comprising: a plurality ofdevice components, each including at least one layer of geometricstructures and having optical properties for a particular opticalresponse, wherein portions of the device component; wherein theplurality of device components are shaped to manipulate light defined ina particular wavelength range based on the shapes and sizes of thegeometric structures.
 13. The apparatus of claim 12, wherein theplurality of device components are configured and arranged to support aplurality of optical modes that include inter-mode and intra-modecoupling, which is mediated by the bouncing of light between thedifferent vertical interfaces of the layers.
 14. The apparatus of claim12, wherein the geometric structures have a geometric shape and sizedefined by wavelength dimensions and having optical properties for aparticular optical response, and wherein each device component is adielectric or semiconducting film or films configured and arranged on aflat substrate.
 15. The apparatus of claim 12, wherein the plurality ofdevice components include a single layer of geometric structures and areconfigured to have optical properties for a particular optical response.16. The apparatus of claim 12, wherein the plurality of devicecomponents include multiple layers of geometric structures and areconfigured to have optical properties for a particular optical response.17. The apparatus of claim 12, wherein the geometric structures areformed of at least one material primarily non-metallic and/or having arefractive index that is greater than two and selected from the groupconsisting of metal, insulating, semiconducting material, and acombination thereof, and are configured to have optical properties for aparticular optical response including controlling at least one of anamplitude and phase of light across a broadband spectrum.
 18. Theapparatus of claim 12, wherein each device component includes at leastone layer of geometric structures formed of two or more differentmaterials primarily non-metallic and/or having a refractive index thatis greater than two, and formed by optimizing a topology and boundaries,for each device component, to have particular optical properties for aparticular optical response and based on formation constraints.
 19. Theapparatus of claim 12, wherein each device component includes at leastone layer of geometric structures and has a particular optical responseas a function of at least one of: a number of layers of geometricstructures of the device component, dimensions of the device component,thickness of each layer of the device component, materials forming thegeometric structures, presence of a layer of solid material, and totalthickness of the device component.
 20. The apparatus of claim 12,wherein the plurality of device components are coupled together to forman aperiodic apparatus, and wherein one or more of the layers ofgeometric structures is formed of two or more materials selected fromthe group consisting of phase change materials, electro-optic materials,and electrochemical materials.